Eliminate a lot of list-management overhead within join_search_one_level

by adding a requirement that build_join_rel add new join RelOptInfos to the
appropriate list immediately at creation.  Per report from Robert Haas,
the list_concat_unique_ptr() calls that this change eliminates were taking
the lion's share of the runtime in larger join problems.  This doesn't do
anything to fix the fundamental combinatorial explosion in large join
problems, but it should push out the threshold of pain a bit further.

Note: because this changes the order in which joinrel lists are built,
it might result in changes in selected plans in cases where different
alternatives have exactly the same costs.  There is one example in the
regression tests.

diff --git a/src/backend/nodes/outfuncs.c b/src/backend/nodes/outfuncs.c
index 4e1c96271a94866fc1162f0035acab0504f7eefb..690967595044daaa82900aa468cef1a8d2012313 100644 (file)
--- a/src/backend/nodes/outfuncs.c
+++ b/src/backend/nodes/outfuncs.c
@@ -8,7 +8,7 @@
  *
  *
  * IDENTIFICATION
- *   $PostgreSQL: pgsql/src/backend/nodes/outfuncs.c,v 1.372 2009/11/15 02:45:34 tgl Exp $
+ *   $PostgreSQL: pgsql/src/backend/nodes/outfuncs.c,v 1.373 2009/11/28 00:46:18 tgl Exp $
  *
  * NOTES
  *   Every node type that can appear in stored rules' parsetrees *must*
@@ -1544,6 +1544,7 @@ _outPlannerInfo(StringInfo str, PlannerInfo *node)
    WRITE_NODE_FIELD(glob);
    WRITE_UINT_FIELD(query_level);
    WRITE_NODE_FIELD(join_rel_list);
+   WRITE_INT_FIELD(join_cur_level);
    WRITE_NODE_FIELD(resultRelations);
    WRITE_NODE_FIELD(init_plans);
    WRITE_NODE_FIELD(cte_plan_ids);

diff --git a/src/backend/optimizer/geqo/geqo_eval.c b/src/backend/optimizer/geqo/geqo_eval.c
index ea74abdde5e0bb0a0434d0ddd9755670022a342a..1e6428e5871e5d4fc43d606de2c68ad2624bdba1 100644 (file)
--- a/src/backend/optimizer/geqo/geqo_eval.c
+++ b/src/backend/optimizer/geqo/geqo_eval.c
@@ -6,7 +6,7 @@
  * Portions Copyright (c) 1996-2009, PostgreSQL Global Development Group
  * Portions Copyright (c) 1994, Regents of the University of California
  *
- * $PostgreSQL: pgsql/src/backend/optimizer/geqo/geqo_eval.c,v 1.90 2009/07/19 21:00:43 tgl Exp $
+ * $PostgreSQL: pgsql/src/backend/optimizer/geqo/geqo_eval.c,v 1.91 2009/11/28 00:46:18 tgl Exp $
  *
  *-------------------------------------------------------------------------
  */
@@ -88,9 +88,12 @@ geqo_eval(PlannerInfo *root, Gene *tour, int num_gene)
     * is one.  We can do this by just temporarily setting the link to NULL.
     * (If we are dealing with enough join rels, which we very likely are, a
     * new hash table will get built and used locally.)
+    *
+    * join_rel_level[] shouldn't be in use, so just Assert it isn't.
     */
    savelength = list_length(root->join_rel_list);
    savehash = root->join_rel_hash;
+   Assert(root->join_rel_level == NULL);
 
    root->join_rel_hash = NULL;
 

diff --git a/src/backend/optimizer/path/allpaths.c b/src/backend/optimizer/path/allpaths.c
index bdc77662bff55ebc9f90d3b189939513e71b4a71..e7c6fdf40ac666e5e013a70d28c0a117e4d87b64 100644 (file)
--- a/src/backend/optimizer/path/allpaths.c
+++ b/src/backend/optimizer/path/allpaths.c
@@ -8,7 +8,7 @@
  *
  *
  * IDENTIFICATION
- *   $PostgreSQL: pgsql/src/backend/optimizer/path/allpaths.c,v 1.190 2009/11/22 14:54:31 heikki Exp $
+ *   $PostgreSQL: pgsql/src/backend/optimizer/path/allpaths.c,v 1.191 2009/11/28 00:46:19 tgl Exp $
  *
  *-------------------------------------------------------------------------
  */
@@ -898,10 +898,15 @@ make_rel_from_joinlist(PlannerInfo *root, List *joinlist)
 RelOptInfo *
 standard_join_search(PlannerInfo *root, int levels_needed, List *initial_rels)
 {
-   List      **joinitems;
    int         lev;
    RelOptInfo *rel;
 
+   /*
+    * This function cannot be invoked recursively within any one planning
+    * problem, so join_rel_level[] can't be in use already.
+    */
+   Assert(root->join_rel_level == NULL);
+
    /*
     * We employ a simple "dynamic programming" algorithm: we first find all
     * ways to build joins of two jointree items, then all ways to build joins
@@ -909,30 +914,31 @@ standard_join_search(PlannerInfo *root, int levels_needed, List *initial_rels)
     * joins, and so on until we have considered all ways to join all the
     * items into one rel.
     *
-    * joinitems[j] is a list of all the j-item rels.  Initially we set
-    * joinitems[1] to represent all the single-jointree-item relations.
+    * root->join_rel_level[j] is a list of all the j-item rels.  Initially we
+    * set root->join_rel_level[1] to represent all the single-jointree-item
+    * relations.
     */
-   joinitems = (List **) palloc0((levels_needed + 1) * sizeof(List *));
+   root->join_rel_level = (List **) palloc0((levels_needed + 1) * sizeof(List *));
 
-   joinitems[1] = initial_rels;
+   root->join_rel_level[1] = initial_rels;
 
    for (lev = 2; lev <= levels_needed; lev++)
    {
-       ListCell   *x;
+       ListCell   *lc;
 
        /*
         * Determine all possible pairs of relations to be joined at this
         * level, and build paths for making each one from every available
         * pair of lower-level relations.
         */
-       joinitems[lev] = join_search_one_level(root, lev, joinitems);
+       join_search_one_level(root, lev);
 
        /*
         * Do cleanup work on each just-processed rel.
         */
-       foreach(x, joinitems[lev])
+       foreach(lc, root->join_rel_level[lev])
        {
-           rel = (RelOptInfo *) lfirst(x);
+           rel = (RelOptInfo *) lfirst(lc);
 
            /* Find and save the cheapest paths for this rel */
            set_cheapest(rel);
@@ -946,11 +952,13 @@ standard_join_search(PlannerInfo *root, int levels_needed, List *initial_rels)
    /*
     * We should have a single rel at the final level.
     */
-   if (joinitems[levels_needed] == NIL)
+   if (root->join_rel_level[levels_needed] == NIL)
        elog(ERROR, "failed to build any %d-way joins", levels_needed);
-   Assert(list_length(joinitems[levels_needed]) == 1);
+   Assert(list_length(root->join_rel_level[levels_needed]) == 1);
+
+   rel = (RelOptInfo *) linitial(root->join_rel_level[levels_needed]);
 
-   rel = (RelOptInfo *) linitial(joinitems[levels_needed]);
+   root->join_rel_level = NULL;
 
    return rel;
 }

diff --git a/src/backend/optimizer/path/joinrels.c b/src/backend/optimizer/path/joinrels.c
index 49e8a1222aa2351f0b6fc97a0319116b1bfb624b..ee43326f13e2c2f3dafc9458dc3591f6e8318e4e 100644 (file)
--- a/src/backend/optimizer/path/joinrels.c
+++ b/src/backend/optimizer/path/joinrels.c
@@ -8,7 +8,7 @@
  *
  *
  * IDENTIFICATION
- *   $PostgreSQL: pgsql/src/backend/optimizer/path/joinrels.c,v 1.102 2009/07/23 17:42:06 tgl Exp $
+ *   $PostgreSQL: pgsql/src/backend/optimizer/path/joinrels.c,v 1.103 2009/11/28 00:46:19 tgl Exp $
  *
  *-------------------------------------------------------------------------
  */
@@ -19,10 +19,10 @@
 #include "optimizer/paths.h"
 
 
-static List *make_rels_by_clause_joins(PlannerInfo *root,
+static void make_rels_by_clause_joins(PlannerInfo *root,
                          RelOptInfo *old_rel,
                          ListCell *other_rels);
-static List *make_rels_by_clauseless_joins(PlannerInfo *root,
+static void make_rels_by_clauseless_joins(PlannerInfo *root,
                              RelOptInfo *old_rel,
                              ListCell *other_rels);
 static bool has_join_restriction(PlannerInfo *root, RelOptInfo *rel);
@@ -40,17 +40,23 @@ static bool restriction_is_constant_false(List *restrictlist);
  *   combination of lower-level rels are created and returned in a list.
  *   Implementation paths are created for each such joinrel, too.
  *
- * level: level of rels we want to make this time.
- * joinrels[j], 1 <= j < level, is a list of rels containing j items.
+ * level: level of rels we want to make this time
+ * root->join_rel_level[j], 1 <= j < level, is a list of rels containing j items
+ *
+ * The result is returned in root->join_rel_level[level].
  */
-List *
-join_search_one_level(PlannerInfo *root, int level, List **joinrels)
+void
+join_search_one_level(PlannerInfo *root, int level)
 {
-   List       *result_rels = NIL;
-   List       *new_rels;
+   List      **joinrels = root->join_rel_level;
    ListCell   *r;
    int         k;
 
+   Assert(joinrels[level] == NIL);
+
+   /* Set join_cur_level so that new joinrels are added to proper list */
+   root->join_cur_level = level;
+
    /*
     * First, consider left-sided and right-sided plans, in which rels of
     * exactly level-1 member relations are joined against initial relations.
@@ -88,9 +94,9 @@ join_search_one_level(PlannerInfo *root, int level, List **joinrels)
             *
             * See also the last-ditch case below.
             */
-           new_rels = make_rels_by_clause_joins(root,
-                                                old_rel,
-                                                other_rels);
+           make_rels_by_clause_joins(root,
+                                     old_rel,
+                                     other_rels);
        }
        else
        {
@@ -99,20 +105,10 @@ join_search_one_level(PlannerInfo *root, int level, List **joinrels)
             * relation, either directly or by join-order restrictions.
             * Cartesian product time.
             */
-           new_rels = make_rels_by_clauseless_joins(root,
-                                                    old_rel,
-                                                    other_rels);
+           make_rels_by_clauseless_joins(root,
+                                         old_rel,
+                                         other_rels);
        }
-
-       /*
-        * At levels above 2 we will generate the same joined relation in
-        * multiple ways --- for example (a join b) join c is the same
-        * RelOptInfo as (b join c) join a, though the second case will add a
-        * different set of Paths to it.  To avoid making extra work for
-        * subsequent passes, do not enter the same RelOptInfo into our output
-        * list multiple times.
-        */
-       result_rels = list_concat_unique_ptr(result_rels, new_rels);
    }
 
    /*
@@ -168,13 +164,7 @@ join_search_one_level(PlannerInfo *root, int level, List **joinrels)
                    if (have_relevant_joinclause(root, old_rel, new_rel) ||
                        have_join_order_restriction(root, old_rel, new_rel))
                    {
-                       RelOptInfo *jrel;
-
-                       jrel = make_join_rel(root, old_rel, new_rel);
-                       /* Avoid making duplicate entries ... */
-                       if (jrel)
-                           result_rels = list_append_unique_ptr(result_rels,
-                                                                jrel);
+                       (void) make_join_rel(root, old_rel, new_rel);
                    }
                }
            }
@@ -193,7 +183,7 @@ join_search_one_level(PlannerInfo *root, int level, List **joinrels)
     * choice but to make cartesian joins.  We consider only left-sided and
     * right-sided cartesian joins in this case (no bushy).
     */
-   if (result_rels == NIL)
+   if (joinrels[level] == NIL)
    {
        /*
         * This loop is just like the first one, except we always call
@@ -211,11 +201,9 @@ join_search_one_level(PlannerInfo *root, int level, List **joinrels)
                other_rels = list_head(joinrels[1]);    /* consider all initial
                                                         * rels */
 
-           new_rels = make_rels_by_clauseless_joins(root,
-                                                    old_rel,
-                                                    other_rels);
-
-           result_rels = list_concat_unique_ptr(result_rels, new_rels);
+           make_rels_by_clauseless_joins(root,
+                                         old_rel,
+                                         other_rels);
        }
 
        /*----------
@@ -235,11 +223,9 @@ join_search_one_level(PlannerInfo *root, int level, List **joinrels)
         * never fail, and so the following sanity check is useful.
         *----------
         */
-       if (result_rels == NIL && root->join_info_list == NIL)
+       if (joinrels[level] == NIL && root->join_info_list == NIL)
            elog(ERROR, "failed to build any %d-way joins", level);
    }
-
-   return result_rels;
 }
 
 /*
@@ -247,7 +233,13 @@ join_search_one_level(PlannerInfo *root, int level, List **joinrels)
  *   Build joins between the given relation 'old_rel' and other relations
  *   that participate in join clauses that 'old_rel' also participates in
  *   (or participate in join-order restrictions with it).
- *   The join rel nodes are returned in a list.
+ *   The join rels are returned in root->join_rel_level[join_cur_level].
+ *
+ * Note: at levels above 2 we will generate the same joined relation in
+ * multiple ways --- for example (a join b) join c is the same RelOptInfo as
+ * (b join c) join a, though the second case will add a different set of Paths
+ * to it.  This is the reason for using the join_rel_level mechanism, which
+ * automatically ensures that each new joinrel is only added to the list once.
  *
  * 'old_rel' is the relation entry for the relation to be joined
  * 'other_rels': the first cell in a linked list containing the other
@@ -256,12 +248,11 @@ join_search_one_level(PlannerInfo *root, int level, List **joinrels)
  * Currently, this is only used with initial rels in other_rels, but it
  * will work for joining to joinrels too.
  */
-static List *
+static void
 make_rels_by_clause_joins(PlannerInfo *root,
                          RelOptInfo *old_rel,
                          ListCell *other_rels)
 {
-   List       *result = NIL;
    ListCell   *l;
 
    for_each_cell(l, other_rels)
@@ -272,15 +263,9 @@ make_rels_by_clause_joins(PlannerInfo *root,
            (have_relevant_joinclause(root, old_rel, other_rel) ||
             have_join_order_restriction(root, old_rel, other_rel)))
        {
-           RelOptInfo *jrel;
-
-           jrel = make_join_rel(root, old_rel, other_rel);
-           if (jrel)
-               result = lcons(jrel, result);
+           (void) make_join_rel(root, old_rel, other_rel);
        }
    }
-
-   return result;
 }
 
 /*
@@ -288,7 +273,7 @@ make_rels_by_clause_joins(PlannerInfo *root,
  *   Given a relation 'old_rel' and a list of other relations
  *   'other_rels', create a join relation between 'old_rel' and each
  *   member of 'other_rels' that isn't already included in 'old_rel'.
- *   The join rel nodes are returned in a list.
+ *   The join rels are returned in root->join_rel_level[join_cur_level].
  *
  * 'old_rel' is the relation entry for the relation to be joined
  * 'other_rels': the first cell of a linked list containing the
@@ -297,34 +282,22 @@ make_rels_by_clause_joins(PlannerInfo *root,
  * Currently, this is only used with initial rels in other_rels, but it would
  * work for joining to joinrels too.
  */
-static List *
+static void
 make_rels_by_clauseless_joins(PlannerInfo *root,
                              RelOptInfo *old_rel,
                              ListCell *other_rels)
 {
-   List       *result = NIL;
-   ListCell   *i;
+   ListCell   *l;
 
-   for_each_cell(i, other_rels)
+   for_each_cell(l, other_rels)
    {
-       RelOptInfo *other_rel = (RelOptInfo *) lfirst(i);
+       RelOptInfo *other_rel = (RelOptInfo *) lfirst(l);
 
        if (!bms_overlap(other_rel->relids, old_rel->relids))
        {
-           RelOptInfo *jrel;
-
-           jrel = make_join_rel(root, old_rel, other_rel);
-
-           /*
-            * As long as given other_rels are distinct, don't need to test to
-            * see if jrel is already part of output list.
-            */
-           if (jrel)
-               result = lcons(jrel, result);
+           (void) make_join_rel(root, old_rel, other_rel);
        }
    }
-
-   return result;
 }
 
 

diff --git a/src/backend/optimizer/plan/planmain.c b/src/backend/optimizer/plan/planmain.c
index 0b75d150ab969e622d60fad2c939401de52b500a..8492b11551e61e98a7dec64b883b5659ac598c01 100644 (file)
--- a/src/backend/optimizer/plan/planmain.c
+++ b/src/backend/optimizer/plan/planmain.c
@@ -14,7 +14,7 @@
  *
  *
  * IDENTIFICATION
- *   $PostgreSQL: pgsql/src/backend/optimizer/plan/planmain.c,v 1.115 2009/06/11 14:48:59 momjian Exp $
+ *   $PostgreSQL: pgsql/src/backend/optimizer/plan/planmain.c,v 1.116 2009/11/28 00:46:19 tgl Exp $
  *
  *-------------------------------------------------------------------------
  */
@@ -142,6 +142,8 @@ query_planner(PlannerInfo *root, List *tlist,
        palloc0(root->simple_rel_array_size * sizeof(RelOptInfo *));
    root->join_rel_list = NIL;
    root->join_rel_hash = NULL;
+   root->join_rel_level = NULL;
+   root->join_cur_level = 0;
    root->canon_pathkeys = NIL;
    root->left_join_clauses = NIL;
    root->right_join_clauses = NIL;

diff --git a/src/backend/optimizer/util/relnode.c b/src/backend/optimizer/util/relnode.c
index 58505949331c126350fce3c81dbe8b409b5a0945..70c3dc81ab4684aefabc3d8e73f1341eff2b8837 100644 (file)
--- a/src/backend/optimizer/util/relnode.c
+++ b/src/backend/optimizer/util/relnode.c
@@ -8,7 +8,7 @@
  *
  *
  * IDENTIFICATION
- *   $PostgreSQL: pgsql/src/backend/optimizer/util/relnode.c,v 1.95 2009/10/12 18:10:48 tgl Exp $
+ *   $PostgreSQL: pgsql/src/backend/optimizer/util/relnode.c,v 1.96 2009/11/28 00:46:19 tgl Exp $
  *
  *-------------------------------------------------------------------------
  */
@@ -402,6 +402,20 @@ build_join_rel(PlannerInfo *root,
        hentry->join_rel = joinrel;
    }
 
+   /*
+    * Also, if dynamic-programming join search is active, add the new joinrel
+    * to the appropriate sublist.  Note: you might think the Assert on
+    * number of members should be for equality, but some of the level 1
+    * rels might have been joinrels already, so we can only assert <=.
+    */
+   if (root->join_rel_level)
+   {
+       Assert(root->join_cur_level > 0);
+       Assert(root->join_cur_level <= bms_num_members(joinrel->relids));
+       root->join_rel_level[root->join_cur_level] =
+           lappend(root->join_rel_level[root->join_cur_level], joinrel);
+   }
+
    return joinrel;
 }
 

diff --git a/src/include/nodes/relation.h b/src/include/nodes/relation.h
index d54770eabc446ade48cac7e426e62792d0096b73..8738a9924d553d6f9d30a612e1035e54cf24833d 100644 (file)
--- a/src/include/nodes/relation.h
+++ b/src/include/nodes/relation.h
@@ -7,7 +7,7 @@
  * Portions Copyright (c) 1996-2009, PostgreSQL Global Development Group
  * Portions Copyright (c) 1994, Regents of the University of California
  *
- * $PostgreSQL: pgsql/src/include/nodes/relation.h,v 1.179 2009/11/15 02:45:35 tgl Exp $
+ * $PostgreSQL: pgsql/src/include/nodes/relation.h,v 1.180 2009/11/28 00:46:19 tgl Exp $
  *
  *-------------------------------------------------------------------------
  */
@@ -142,6 +142,16 @@ typedef struct PlannerInfo
    List       *join_rel_list;  /* list of join-relation RelOptInfos */
    struct HTAB *join_rel_hash; /* optional hashtable for join relations */
 
+   /*
+    * When doing a dynamic-programming-style join search, join_rel_level[k]
+    * is a list of all join-relation RelOptInfos of level k, and
+    * join_cur_level is the current level.  New join-relation RelOptInfos
+    * are automatically added to the join_rel_level[join_cur_level] list.
+    * join_rel_level is NULL if not in use.
+    */
+   List      **join_rel_level; /* lists of join-relation RelOptInfos */
+   int         join_cur_level; /* index of list being extended */
+
    List       *resultRelations;    /* integer list of RT indexes, or NIL */
 
    List       *init_plans;     /* init SubPlans for query */

diff --git a/src/include/optimizer/paths.h b/src/include/optimizer/paths.h
index d9e91675f2261b4456f762305a4fe1770f08c46a..4d04406d8592ae76b918a36d4552779eb77d4ebf 100644 (file)
--- a/src/include/optimizer/paths.h
+++ b/src/include/optimizer/paths.h
@@ -7,7 +7,7 @@
  * Portions Copyright (c) 1996-2009, PostgreSQL Global Development Group
  * Portions Copyright (c) 1994, Regents of the University of California
  *
- * $PostgreSQL: pgsql/src/include/optimizer/paths.h,v 1.108 2009/09/17 20:49:29 tgl Exp $
+ * $PostgreSQL: pgsql/src/include/optimizer/paths.h,v 1.109 2009/11/28 00:46:19 tgl Exp $
  *
  *-------------------------------------------------------------------------
  */
@@ -99,8 +99,7 @@ extern void add_paths_to_joinrel(PlannerInfo *root, RelOptInfo *joinrel,
  * joinrels.c
  *   routines to determine which relations to join
  */
-extern List *join_search_one_level(PlannerInfo *root, int level,
-                     List **joinrels);
+extern void join_search_one_level(PlannerInfo *root, int level);
 extern RelOptInfo *make_join_rel(PlannerInfo *root,
              RelOptInfo *rel1, RelOptInfo *rel2);
 extern bool have_join_order_restriction(PlannerInfo *root,

diff --git a/src/test/regress/expected/join.out b/src/test/regress/expected/join.out
index 0dc3e56960374cbbe326d10decf961c9281a82f5..fed424cb33b99ae65df93f7fe6ff93a418a51c60 100644 (file)
--- a/src/test/regress/expected/join.out
+++ b/src/test/regress/expected/join.out
@@ -662,895 +662,895 @@ SELECT '' AS "xxx", *
  xxx | i | j |   t   | i | k  | i | k  
 -----+---+---+-------+---+----+---+----
      | 1 | 4 | one   | 1 | -1 | 1 | -1
-     | 1 | 4 | one   | 2 |  2 | 1 | -1
-     | 1 | 4 | one   | 3 | -3 | 1 | -1
-     | 1 | 4 | one   | 2 |  4 | 1 | -1
-     | 1 | 4 | one   | 5 | -5 | 1 | -1
-     | 1 | 4 | one   | 5 | -5 | 1 | -1
-     | 1 | 4 | one   | 0 |    | 1 | -1
-     | 1 | 4 | one   |   |    | 1 | -1
-     | 1 | 4 | one   |   |  0 | 1 | -1
+     | 1 | 4 | one   | 1 | -1 | 2 |  2
+     | 1 | 4 | one   | 1 | -1 | 3 | -3
+     | 1 | 4 | one   | 1 | -1 | 2 |  4
+     | 1 | 4 | one   | 1 | -1 | 5 | -5
+     | 1 | 4 | one   | 1 | -1 | 5 | -5
+     | 1 | 4 | one   | 1 | -1 | 0 |   
+     | 1 | 4 | one   | 1 | -1 |   |   
+     | 1 | 4 | one   | 1 | -1 |   |  0
      | 2 | 3 | two   | 1 | -1 | 1 | -1
-     | 2 | 3 | two   | 2 |  2 | 1 | -1
-     | 2 | 3 | two   | 3 | -3 | 1 | -1
-     | 2 | 3 | two   | 2 |  4 | 1 | -1
-     | 2 | 3 | two   | 5 | -5 | 1 | -1
-     | 2 | 3 | two   | 5 | -5 | 1 | -1
-     | 2 | 3 | two   | 0 |    | 1 | -1
-     | 2 | 3 | two   |   |    | 1 | -1
-     | 2 | 3 | two   |   |  0 | 1 | -1
+     | 2 | 3 | two   | 1 | -1 | 2 |  2
+     | 2 | 3 | two   | 1 | -1 | 3 | -3
+     | 2 | 3 | two   | 1 | -1 | 2 |  4
+     | 2 | 3 | two   | 1 | -1 | 5 | -5
+     | 2 | 3 | two   | 1 | -1 | 5 | -5
+     | 2 | 3 | two   | 1 | -1 | 0 |   
+     | 2 | 3 | two   | 1 | -1 |   |   
+     | 2 | 3 | two   | 1 | -1 |   |  0
      | 3 | 2 | three | 1 | -1 | 1 | -1
-     | 3 | 2 | three | 2 |  2 | 1 | -1
-     | 3 | 2 | three | 3 | -3 | 1 | -1
-     | 3 | 2 | three | 2 |  4 | 1 | -1
-     | 3 | 2 | three | 5 | -5 | 1 | -1
-     | 3 | 2 | three | 5 | -5 | 1 | -1
-     | 3 | 2 | three | 0 |    | 1 | -1
-     | 3 | 2 | three |   |    | 1 | -1
-     | 3 | 2 | three |   |  0 | 1 | -1
+     | 3 | 2 | three | 1 | -1 | 2 |  2
+     | 3 | 2 | three | 1 | -1 | 3 | -3
+     | 3 | 2 | three | 1 | -1 | 2 |  4
+     | 3 | 2 | three | 1 | -1 | 5 | -5
+     | 3 | 2 | three | 1 | -1 | 5 | -5
+     | 3 | 2 | three | 1 | -1 | 0 |   
+     | 3 | 2 | three | 1 | -1 |   |   
+     | 3 | 2 | three | 1 | -1 |   |  0
      | 4 | 1 | four  | 1 | -1 | 1 | -1
-     | 4 | 1 | four  | 2 |  2 | 1 | -1
-     | 4 | 1 | four  | 3 | -3 | 1 | -1
-     | 4 | 1 | four  | 2 |  4 | 1 | -1
-     | 4 | 1 | four  | 5 | -5 | 1 | -1
-     | 4 | 1 | four  | 5 | -5 | 1 | -1
-     | 4 | 1 | four  | 0 |    | 1 | -1
-     | 4 | 1 | four  |   |    | 1 | -1
-     | 4 | 1 | four  |   |  0 | 1 | -1
+     | 4 | 1 | four  | 1 | -1 | 2 |  2
+     | 4 | 1 | four  | 1 | -1 | 3 | -3
+     | 4 | 1 | four  | 1 | -1 | 2 |  4
+     | 4 | 1 | four  | 1 | -1 | 5 | -5
+     | 4 | 1 | four  | 1 | -1 | 5 | -5
+     | 4 | 1 | four  | 1 | -1 | 0 |   
+     | 4 | 1 | four  | 1 | -1 |   |   
+     | 4 | 1 | four  | 1 | -1 |   |  0
      | 5 | 0 | five  | 1 | -1 | 1 | -1
-     | 5 | 0 | five  | 2 |  2 | 1 | -1
-     | 5 | 0 | five  | 3 | -3 | 1 | -1
-     | 5 | 0 | five  | 2 |  4 | 1 | -1
-     | 5 | 0 | five  | 5 | -5 | 1 | -1
-     | 5 | 0 | five  | 5 | -5 | 1 | -1
-     | 5 | 0 | five  | 0 |    | 1 | -1
-     | 5 | 0 | five  |   |    | 1 | -1
-     | 5 | 0 | five  |   |  0 | 1 | -1
+     | 5 | 0 | five  | 1 | -1 | 2 |  2
+     | 5 | 0 | five  | 1 | -1 | 3 | -3
+     | 5 | 0 | five  | 1 | -1 | 2 |  4
+     | 5 | 0 | five  | 1 | -1 | 5 | -5
+     | 5 | 0 | five  | 1 | -1 | 5 | -5
+     | 5 | 0 | five  | 1 | -1 | 0 |   
+     | 5 | 0 | five  | 1 | -1 |   |   
+     | 5 | 0 | five  | 1 | -1 |   |  0
      | 6 | 6 | six   | 1 | -1 | 1 | -1
-     | 6 | 6 | six   | 2 |  2 | 1 | -1
-     | 6 | 6 | six   | 3 | -3 | 1 | -1
-     | 6 | 6 | six   | 2 |  4 | 1 | -1
-     | 6 | 6 | six   | 5 | -5 | 1 | -1
-     | 6 | 6 | six   | 5 | -5 | 1 | -1
-     | 6 | 6 | six   | 0 |    | 1 | -1
-     | 6 | 6 | six   |   |    | 1 | -1
-     | 6 | 6 | six   |   |  0 | 1 | -1
+     | 6 | 6 | six   | 1 | -1 | 2 |  2
+     | 6 | 6 | six   | 1 | -1 | 3 | -3
+     | 6 | 6 | six   | 1 | -1 | 2 |  4
+     | 6 | 6 | six   | 1 | -1 | 5 | -5
+     | 6 | 6 | six   | 1 | -1 | 5 | -5
+     | 6 | 6 | six   | 1 | -1 | 0 |   
+     | 6 | 6 | six   | 1 | -1 |   |   
+     | 6 | 6 | six   | 1 | -1 |   |  0
      | 7 | 7 | seven | 1 | -1 | 1 | -1
-     | 7 | 7 | seven | 2 |  2 | 1 | -1
-     | 7 | 7 | seven | 3 | -3 | 1 | -1
-     | 7 | 7 | seven | 2 |  4 | 1 | -1
-     | 7 | 7 | seven | 5 | -5 | 1 | -1
-     | 7 | 7 | seven | 5 | -5 | 1 | -1
-     | 7 | 7 | seven | 0 |    | 1 | -1
-     | 7 | 7 | seven |   |    | 1 | -1
-     | 7 | 7 | seven |   |  0 | 1 | -1
+     | 7 | 7 | seven | 1 | -1 | 2 |  2
+     | 7 | 7 | seven | 1 | -1 | 3 | -3
+     | 7 | 7 | seven | 1 | -1 | 2 |  4
+     | 7 | 7 | seven | 1 | -1 | 5 | -5
+     | 7 | 7 | seven | 1 | -1 | 5 | -5
+     | 7 | 7 | seven | 1 | -1 | 0 |   
+     | 7 | 7 | seven | 1 | -1 |   |   
+     | 7 | 7 | seven | 1 | -1 |   |  0
      | 8 | 8 | eight | 1 | -1 | 1 | -1
-     | 8 | 8 | eight | 2 |  2 | 1 | -1
-     | 8 | 8 | eight | 3 | -3 | 1 | -1
-     | 8 | 8 | eight | 2 |  4 | 1 | -1
-     | 8 | 8 | eight | 5 | -5 | 1 | -1
-     | 8 | 8 | eight | 5 | -5 | 1 | -1
-     | 8 | 8 | eight | 0 |    | 1 | -1
-     | 8 | 8 | eight |   |    | 1 | -1
-     | 8 | 8 | eight |   |  0 | 1 | -1
+     | 8 | 8 | eight | 1 | -1 | 2 |  2
+     | 8 | 8 | eight | 1 | -1 | 3 | -3
+     | 8 | 8 | eight | 1 | -1 | 2 |  4
+     | 8 | 8 | eight | 1 | -1 | 5 | -5
+     | 8 | 8 | eight | 1 | -1 | 5 | -5
+     | 8 | 8 | eight | 1 | -1 | 0 |   
+     | 8 | 8 | eight | 1 | -1 |   |   
+     | 8 | 8 | eight | 1 | -1 |   |  0
      | 0 |   | zero  | 1 | -1 | 1 | -1
-     | 0 |   | zero  | 2 |  2 | 1 | -1
-     | 0 |   | zero  | 3 | -3 | 1 | -1
-     | 0 |   | zero  | 2 |  4 | 1 | -1
-     | 0 |   | zero  | 5 | -5 | 1 | -1
-     | 0 |   | zero  | 5 | -5 | 1 | -1
-     | 0 |   | zero  | 0 |    | 1 | -1
-     | 0 |   | zero  |   |    | 1 | -1
-     | 0 |   | zero  |   |  0 | 1 | -1
+     | 0 |   | zero  | 1 | -1 | 2 |  2
+     | 0 |   | zero  | 1 | -1 | 3 | -3
+     | 0 |   | zero  | 1 | -1 | 2 |  4
+     | 0 |   | zero  | 1 | -1 | 5 | -5
+     | 0 |   | zero  | 1 | -1 | 5 | -5
+     | 0 |   | zero  | 1 | -1 | 0 |   
+     | 0 |   | zero  | 1 | -1 |   |   
+     | 0 |   | zero  | 1 | -1 |   |  0
      |   |   | null  | 1 | -1 | 1 | -1
-     |   |   | null  | 2 |  2 | 1 | -1
-     |   |   | null  | 3 | -3 | 1 | -1
-     |   |   | null  | 2 |  4 | 1 | -1
-     |   |   | null  | 5 | -5 | 1 | -1
-     |   |   | null  | 5 | -5 | 1 | -1
-     |   |   | null  | 0 |    | 1 | -1
-     |   |   | null  |   |    | 1 | -1
-     |   |   | null  |   |  0 | 1 | -1
+     |   |   | null  | 1 | -1 | 2 |  2
+     |   |   | null  | 1 | -1 | 3 | -3
+     |   |   | null  | 1 | -1 | 2 |  4
+     |   |   | null  | 1 | -1 | 5 | -5
+     |   |   | null  | 1 | -1 | 5 | -5
+     |   |   | null  | 1 | -1 | 0 |   
+     |   |   | null  | 1 | -1 |   |   
+     |   |   | null  | 1 | -1 |   |  0
      |   | 0 | zero  | 1 | -1 | 1 | -1
-     |   | 0 | zero  | 2 |  2 | 1 | -1
-     |   | 0 | zero  | 3 | -3 | 1 | -1
-     |   | 0 | zero  | 2 |  4 | 1 | -1
-     |   | 0 | zero  | 5 | -5 | 1 | -1
-     |   | 0 | zero  | 5 | -5 | 1 | -1
-     |   | 0 | zero  | 0 |    | 1 | -1
-     |   | 0 | zero  |   |    | 1 | -1
-     |   | 0 | zero  |   |  0 | 1 | -1
-     | 1 | 4 | one   | 1 | -1 | 2 |  2
+     |   | 0 | zero  | 1 | -1 | 2 |  2
+     |   | 0 | zero  | 1 | -1 | 3 | -3
+     |   | 0 | zero  | 1 | -1 | 2 |  4
+     |   | 0 | zero  | 1 | -1 | 5 | -5
+     |   | 0 | zero  | 1 | -1 | 5 | -5
+     |   | 0 | zero  | 1 | -1 | 0 |   
+     |   | 0 | zero  | 1 | -1 |   |   
+     |   | 0 | zero  | 1 | -1 |   |  0
+     | 1 | 4 | one   | 2 |  2 | 1 | -1
      | 1 | 4 | one   | 2 |  2 | 2 |  2
-     | 1 | 4 | one   | 3 | -3 | 2 |  2
-     | 1 | 4 | one   | 2 |  4 | 2 |  2
-     | 1 | 4 | one   | 5 | -5 | 2 |  2
-     | 1 | 4 | one   | 5 | -5 | 2 |  2
-     | 1 | 4 | one   | 0 |    | 2 |  2
-     | 1 | 4 | one   |   |    | 2 |  2
-     | 1 | 4 | one   |   |  0 | 2 |  2
-     | 2 | 3 | two   | 1 | -1 | 2 |  2
+     | 1 | 4 | one   | 2 |  2 | 3 | -3
+     | 1 | 4 | one   | 2 |  2 | 2 |  4
+     | 1 | 4 | one   | 2 |  2 | 5 | -5
+     | 1 | 4 | one   | 2 |  2 | 5 | -5
+     | 1 | 4 | one   | 2 |  2 | 0 |   
+     | 1 | 4 | one   | 2 |  2 |   |   
+     | 1 | 4 | one   | 2 |  2 |   |  0
+     | 2 | 3 | two   | 2 |  2 | 1 | -1
      | 2 | 3 | two   | 2 |  2 | 2 |  2
-     | 2 | 3 | two   | 3 | -3 | 2 |  2
-     | 2 | 3 | two   | 2 |  4 | 2 |  2
-     | 2 | 3 | two   | 5 | -5 | 2 |  2
-     | 2 | 3 | two   | 5 | -5 | 2 |  2
-     | 2 | 3 | two   | 0 |    | 2 |  2
-     | 2 | 3 | two   |   |    | 2 |  2
-     | 2 | 3 | two   |   |  0 | 2 |  2
-     | 3 | 2 | three | 1 | -1 | 2 |  2
+     | 2 | 3 | two   | 2 |  2 | 3 | -3
+     | 2 | 3 | two   | 2 |  2 | 2 |  4
+     | 2 | 3 | two   | 2 |  2 | 5 | -5
+     | 2 | 3 | two   | 2 |  2 | 5 | -5
+     | 2 | 3 | two   | 2 |  2 | 0 |   
+     | 2 | 3 | two   | 2 |  2 |   |   
+     | 2 | 3 | two   | 2 |  2 |   |  0
+     | 3 | 2 | three | 2 |  2 | 1 | -1
      | 3 | 2 | three | 2 |  2 | 2 |  2
-     | 3 | 2 | three | 3 | -3 | 2 |  2
-     | 3 | 2 | three | 2 |  4 | 2 |  2
-     | 3 | 2 | three | 5 | -5 | 2 |  2
-     | 3 | 2 | three | 5 | -5 | 2 |  2
-     | 3 | 2 | three | 0 |    | 2 |  2
-     | 3 | 2 | three |   |    | 2 |  2
-     | 3 | 2 | three |   |  0 | 2 |  2
-     | 4 | 1 | four  | 1 | -1 | 2 |  2
+     | 3 | 2 | three | 2 |  2 | 3 | -3
+     | 3 | 2 | three | 2 |  2 | 2 |  4
+     | 3 | 2 | three | 2 |  2 | 5 | -5
+     | 3 | 2 | three | 2 |  2 | 5 | -5
+     | 3 | 2 | three | 2 |  2 | 0 |   
+     | 3 | 2 | three | 2 |  2 |   |   
+     | 3 | 2 | three | 2 |  2 |   |  0
+     | 4 | 1 | four  | 2 |  2 | 1 | -1
      | 4 | 1 | four  | 2 |  2 | 2 |  2
-     | 4 | 1 | four  | 3 | -3 | 2 |  2
-     | 4 | 1 | four  | 2 |  4 | 2 |  2
-     | 4 | 1 | four  | 5 | -5 | 2 |  2
-     | 4 | 1 | four  | 5 | -5 | 2 |  2
-     | 4 | 1 | four  | 0 |    | 2 |  2
-     | 4 | 1 | four  |   |    | 2 |  2
-     | 4 | 1 | four  |   |  0 | 2 |  2
-     | 5 | 0 | five  | 1 | -1 | 2 |  2
+     | 4 | 1 | four  | 2 |  2 | 3 | -3
+     | 4 | 1 | four  | 2 |  2 | 2 |  4
+     | 4 | 1 | four  | 2 |  2 | 5 | -5
+     | 4 | 1 | four  | 2 |  2 | 5 | -5
+     | 4 | 1 | four  | 2 |  2 | 0 |   
+     | 4 | 1 | four  | 2 |  2 |   |   
+     | 4 | 1 | four  | 2 |  2 |   |  0
+     | 5 | 0 | five  | 2 |  2 | 1 | -1
      | 5 | 0 | five  | 2 |  2 | 2 |  2
-     | 5 | 0 | five  | 3 | -3 | 2 |  2
-     | 5 | 0 | five  | 2 |  4 | 2 |  2
-     | 5 | 0 | five  | 5 | -5 | 2 |  2
-     | 5 | 0 | five  | 5 | -5 | 2 |  2
-     | 5 | 0 | five  | 0 |    | 2 |  2
-     | 5 | 0 | five  |   |    | 2 |  2
-     | 5 | 0 | five  |   |  0 | 2 |  2
-     | 6 | 6 | six   | 1 | -1 | 2 |  2
+     | 5 | 0 | five  | 2 |  2 | 3 | -3
+     | 5 | 0 | five  | 2 |  2 | 2 |  4
+     | 5 | 0 | five  | 2 |  2 | 5 | -5
+     | 5 | 0 | five  | 2 |  2 | 5 | -5
+     | 5 | 0 | five  | 2 |  2 | 0 |   
+     | 5 | 0 | five  | 2 |  2 |   |   
+     | 5 | 0 | five  | 2 |  2 |   |  0
+     | 6 | 6 | six   | 2 |  2 | 1 | -1
      | 6 | 6 | six   | 2 |  2 | 2 |  2
-     | 6 | 6 | six   | 3 | -3 | 2 |  2
-     | 6 | 6 | six   | 2 |  4 | 2 |  2
-     | 6 | 6 | six   | 5 | -5 | 2 |  2
-     | 6 | 6 | six   | 5 | -5 | 2 |  2
-     | 6 | 6 | six   | 0 |    | 2 |  2
-     | 6 | 6 | six   |   |    | 2 |  2
-     | 6 | 6 | six   |   |  0 | 2 |  2
-     | 7 | 7 | seven | 1 | -1 | 2 |  2
+     | 6 | 6 | six   | 2 |  2 | 3 | -3
+     | 6 | 6 | six   | 2 |  2 | 2 |  4
+     | 6 | 6 | six   | 2 |  2 | 5 | -5
+     | 6 | 6 | six   | 2 |  2 | 5 | -5
+     | 6 | 6 | six   | 2 |  2 | 0 |   
+     | 6 | 6 | six   | 2 |  2 |   |   
+     | 6 | 6 | six   | 2 |  2 |   |  0
+     | 7 | 7 | seven | 2 |  2 | 1 | -1
      | 7 | 7 | seven | 2 |  2 | 2 |  2
-     | 7 | 7 | seven | 3 | -3 | 2 |  2
-     | 7 | 7 | seven | 2 |  4 | 2 |  2
-     | 7 | 7 | seven | 5 | -5 | 2 |  2
-     | 7 | 7 | seven | 5 | -5 | 2 |  2
-     | 7 | 7 | seven | 0 |    | 2 |  2
-     | 7 | 7 | seven |   |    | 2 |  2
-     | 7 | 7 | seven |   |  0 | 2 |  2
-     | 8 | 8 | eight | 1 | -1 | 2 |  2
+     | 7 | 7 | seven | 2 |  2 | 3 | -3
+     | 7 | 7 | seven | 2 |  2 | 2 |  4
+     | 7 | 7 | seven | 2 |  2 | 5 | -5
+     | 7 | 7 | seven | 2 |  2 | 5 | -5
+     | 7 | 7 | seven | 2 |  2 | 0 |   
+     | 7 | 7 | seven | 2 |  2 |   |   
+     | 7 | 7 | seven | 2 |  2 |   |  0
+     | 8 | 8 | eight | 2 |  2 | 1 | -1
      | 8 | 8 | eight | 2 |  2 | 2 |  2
-     | 8 | 8 | eight | 3 | -3 | 2 |  2
-     | 8 | 8 | eight | 2 |  4 | 2 |  2
-     | 8 | 8 | eight | 5 | -5 | 2 |  2
-     | 8 | 8 | eight | 5 | -5 | 2 |  2
-     | 8 | 8 | eight | 0 |    | 2 |  2
-     | 8 | 8 | eight |   |    | 2 |  2
-     | 8 | 8 | eight |   |  0 | 2 |  2
-     | 0 |   | zero  | 1 | -1 | 2 |  2
+     | 8 | 8 | eight | 2 |  2 | 3 | -3
+     | 8 | 8 | eight | 2 |  2 | 2 |  4
+     | 8 | 8 | eight | 2 |  2 | 5 | -5
+     | 8 | 8 | eight | 2 |  2 | 5 | -5
+     | 8 | 8 | eight | 2 |  2 | 0 |   
+     | 8 | 8 | eight | 2 |  2 |   |   
+     | 8 | 8 | eight | 2 |  2 |   |  0
+     | 0 |   | zero  | 2 |  2 | 1 | -1
      | 0 |   | zero  | 2 |  2 | 2 |  2
-     | 0 |   | zero  | 3 | -3 | 2 |  2
-     | 0 |   | zero  | 2 |  4 | 2 |  2
-     | 0 |   | zero  | 5 | -5 | 2 |  2
-     | 0 |   | zero  | 5 | -5 | 2 |  2
-     | 0 |   | zero  | 0 |    | 2 |  2
-     | 0 |   | zero  |   |    | 2 |  2
-     | 0 |   | zero  |   |  0 | 2 |  2
-     |   |   | null  | 1 | -1 | 2 |  2
+     | 0 |   | zero  | 2 |  2 | 3 | -3
+     | 0 |   | zero  | 2 |  2 | 2 |  4
+     | 0 |   | zero  | 2 |  2 | 5 | -5
+     | 0 |   | zero  | 2 |  2 | 5 | -5
+     | 0 |   | zero  | 2 |  2 | 0 |   
+     | 0 |   | zero  | 2 |  2 |   |   
+     | 0 |   | zero  | 2 |  2 |   |  0
+     |   |   | null  | 2 |  2 | 1 | -1
      |   |   | null  | 2 |  2 | 2 |  2
-     |   |   | null  | 3 | -3 | 2 |  2
-     |   |   | null  | 2 |  4 | 2 |  2
-     |   |   | null  | 5 | -5 | 2 |  2
-     |   |   | null  | 5 | -5 | 2 |  2
-     |   |   | null  | 0 |    | 2 |  2
-     |   |   | null  |   |    | 2 |  2
-     |   |   | null  |   |  0 | 2 |  2
-     |   | 0 | zero  | 1 | -1 | 2 |  2
+     |   |   | null  | 2 |  2 | 3 | -3
+     |   |   | null  | 2 |  2 | 2 |  4
+     |   |   | null  | 2 |  2 | 5 | -5
+     |   |   | null  | 2 |  2 | 5 | -5
+     |   |   | null  | 2 |  2 | 0 |   
+     |   |   | null  | 2 |  2 |   |   
+     |   |   | null  | 2 |  2 |   |  0
+     |   | 0 | zero  | 2 |  2 | 1 | -1
      |   | 0 | zero  | 2 |  2 | 2 |  2
-     |   | 0 | zero  | 3 | -3 | 2 |  2
-     |   | 0 | zero  | 2 |  4 | 2 |  2
-     |   | 0 | zero  | 5 | -5 | 2 |  2
-     |   | 0 | zero  | 5 | -5 | 2 |  2
-     |   | 0 | zero  | 0 |    | 2 |  2
-     |   | 0 | zero  |   |    | 2 |  2
-     |   | 0 | zero  |   |  0 | 2 |  2
-     | 1 | 4 | one   | 1 | -1 | 3 | -3
-     | 1 | 4 | one   | 2 |  2 | 3 | -3
+     |   | 0 | zero  | 2 |  2 | 3 | -3
+     |   | 0 | zero  | 2 |  2 | 2 |  4
+     |   | 0 | zero  | 2 |  2 | 5 | -5
+     |   | 0 | zero  | 2 |  2 | 5 | -5
+     |   | 0 | zero  | 2 |  2 | 0 |   
+     |   | 0 | zero  | 2 |  2 |   |   
+     |   | 0 | zero  | 2 |  2 |   |  0
+     | 1 | 4 | one   | 3 | -3 | 1 | -1
+     | 1 | 4 | one   | 3 | -3 | 2 |  2
      | 1 | 4 | one   | 3 | -3 | 3 | -3
-     | 1 | 4 | one   | 2 |  4 | 3 | -3
-     | 1 | 4 | one   | 5 | -5 | 3 | -3
-     | 1 | 4 | one   | 5 | -5 | 3 | -3
-     | 1 | 4 | one   | 0 |    | 3 | -3
-     | 1 | 4 | one   |   |    | 3 | -3
-     | 1 | 4 | one   |   |  0 | 3 | -3
-     | 2 | 3 | two   | 1 | -1 | 3 | -3
-     | 2 | 3 | two   | 2 |  2 | 3 | -3
+     | 1 | 4 | one   | 3 | -3 | 2 |  4
+     | 1 | 4 | one   | 3 | -3 | 5 | -5
+     | 1 | 4 | one   | 3 | -3 | 5 | -5
+     | 1 | 4 | one   | 3 | -3 | 0 |   
+     | 1 | 4 | one   | 3 | -3 |   |   
+     | 1 | 4 | one   | 3 | -3 |   |  0
+     | 2 | 3 | two   | 3 | -3 | 1 | -1
+     | 2 | 3 | two   | 3 | -3 | 2 |  2
      | 2 | 3 | two   | 3 | -3 | 3 | -3
-     | 2 | 3 | two   | 2 |  4 | 3 | -3
-     | 2 | 3 | two   | 5 | -5 | 3 | -3
-     | 2 | 3 | two   | 5 | -5 | 3 | -3
-     | 2 | 3 | two   | 0 |    | 3 | -3
-     | 2 | 3 | two   |   |    | 3 | -3
-     | 2 | 3 | two   |   |  0 | 3 | -3
-     | 3 | 2 | three | 1 | -1 | 3 | -3
-     | 3 | 2 | three | 2 |  2 | 3 | -3
+     | 2 | 3 | two   | 3 | -3 | 2 |  4
+     | 2 | 3 | two   | 3 | -3 | 5 | -5
+     | 2 | 3 | two   | 3 | -3 | 5 | -5
+     | 2 | 3 | two   | 3 | -3 | 0 |   
+     | 2 | 3 | two   | 3 | -3 |   |   
+     | 2 | 3 | two   | 3 | -3 |   |  0
+     | 3 | 2 | three | 3 | -3 | 1 | -1
+     | 3 | 2 | three | 3 | -3 | 2 |  2
      | 3 | 2 | three | 3 | -3 | 3 | -3
-     | 3 | 2 | three | 2 |  4 | 3 | -3
-     | 3 | 2 | three | 5 | -5 | 3 | -3
-     | 3 | 2 | three | 5 | -5 | 3 | -3
-     | 3 | 2 | three | 0 |    | 3 | -3
-     | 3 | 2 | three |   |    | 3 | -3
-     | 3 | 2 | three |   |  0 | 3 | -3
-     | 4 | 1 | four  | 1 | -1 | 3 | -3
-     | 4 | 1 | four  | 2 |  2 | 3 | -3
+     | 3 | 2 | three | 3 | -3 | 2 |  4
+     | 3 | 2 | three | 3 | -3 | 5 | -5
+     | 3 | 2 | three | 3 | -3 | 5 | -5
+     | 3 | 2 | three | 3 | -3 | 0 |   
+     | 3 | 2 | three | 3 | -3 |   |   
+     | 3 | 2 | three | 3 | -3 |   |  0
+     | 4 | 1 | four  | 3 | -3 | 1 | -1
+     | 4 | 1 | four  | 3 | -3 | 2 |  2
      | 4 | 1 | four  | 3 | -3 | 3 | -3
-     | 4 | 1 | four  | 2 |  4 | 3 | -3
-     | 4 | 1 | four  | 5 | -5 | 3 | -3
-     | 4 | 1 | four  | 5 | -5 | 3 | -3
-     | 4 | 1 | four  | 0 |    | 3 | -3
-     | 4 | 1 | four  |   |    | 3 | -3
-     | 4 | 1 | four  |   |  0 | 3 | -3
-     | 5 | 0 | five  | 1 | -1 | 3 | -3
-     | 5 | 0 | five  | 2 |  2 | 3 | -3
+     | 4 | 1 | four  | 3 | -3 | 2 |  4
+     | 4 | 1 | four  | 3 | -3 | 5 | -5
+     | 4 | 1 | four  | 3 | -3 | 5 | -5
+     | 4 | 1 | four  | 3 | -3 | 0 |   
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+     | 4 | 1 | four  | 3 | -3 |   |  0
+     | 5 | 0 | five  | 3 | -3 | 1 | -1
+     | 5 | 0 | five  | 3 | -3 | 2 |  2
      | 5 | 0 | five  | 3 | -3 | 3 | -3
-     | 5 | 0 | five  | 2 |  4 | 3 | -3
-     | 5 | 0 | five  | 5 | -5 | 3 | -3
-     | 5 | 0 | five  | 5 | -5 | 3 | -3
-     | 5 | 0 | five  | 0 |    | 3 | -3
-     | 5 | 0 | five  |   |    | 3 | -3
-     | 5 | 0 | five  |   |  0 | 3 | -3
-     | 6 | 6 | six   | 1 | -1 | 3 | -3
-     | 6 | 6 | six   | 2 |  2 | 3 | -3
+     | 5 | 0 | five  | 3 | -3 | 2 |  4
+     | 5 | 0 | five  | 3 | -3 | 5 | -5
+     | 5 | 0 | five  | 3 | -3 | 5 | -5
+     | 5 | 0 | five  | 3 | -3 | 0 |   
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+     | 5 | 0 | five  | 3 | -3 |   |  0
+     | 6 | 6 | six   | 3 | -3 | 1 | -1
+     | 6 | 6 | six   | 3 | -3 | 2 |  2
      | 6 | 6 | six   | 3 | -3 | 3 | -3
-     | 6 | 6 | six   | 2 |  4 | 3 | -3
-     | 6 | 6 | six   | 5 | -5 | 3 | -3
-     | 6 | 6 | six   | 5 | -5 | 3 | -3
-     | 6 | 6 | six   | 0 |    | 3 | -3
-     | 6 | 6 | six   |   |    | 3 | -3
-     | 6 | 6 | six   |   |  0 | 3 | -3
-     | 7 | 7 | seven | 1 | -1 | 3 | -3
-     | 7 | 7 | seven | 2 |  2 | 3 | -3
+     | 6 | 6 | six   | 3 | -3 | 2 |  4
+     | 6 | 6 | six   | 3 | -3 | 5 | -5
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+     | 6 | 6 | six   | 3 | -3 | 0 |   
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+     | 7 | 7 | seven | 3 | -3 | 1 | -1
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      | 7 | 7 | seven | 3 | -3 | 3 | -3
-     | 7 | 7 | seven | 2 |  4 | 3 | -3
-     | 7 | 7 | seven | 5 | -5 | 3 | -3
-     | 7 | 7 | seven | 5 | -5 | 3 | -3
-     | 7 | 7 | seven | 0 |    | 3 | -3
-     | 7 | 7 | seven |   |    | 3 | -3
-     | 7 | 7 | seven |   |  0 | 3 | -3
-     | 8 | 8 | eight | 1 | -1 | 3 | -3
-     | 8 | 8 | eight | 2 |  2 | 3 | -3
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+     | 7 | 7 | seven | 3 | -3 | 5 | -5
+     | 7 | 7 | seven | 3 | -3 | 5 | -5
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+     | 8 | 8 | eight | 3 | -3 | 1 | -1
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      | 8 | 8 | eight | 3 | -3 | 3 | -3
-     | 8 | 8 | eight | 2 |  4 | 3 | -3
-     | 8 | 8 | eight | 5 | -5 | 3 | -3
-     | 8 | 8 | eight | 5 | -5 | 3 | -3
-     | 8 | 8 | eight | 0 |    | 3 | -3
-     | 8 | 8 | eight |   |    | 3 | -3
-     | 8 | 8 | eight |   |  0 | 3 | -3
-     | 0 |   | zero  | 1 | -1 | 3 | -3
-     | 0 |   | zero  | 2 |  2 | 3 | -3
+     | 8 | 8 | eight | 3 | -3 | 2 |  4
+     | 8 | 8 | eight | 3 | -3 | 5 | -5
+     | 8 | 8 | eight | 3 | -3 | 5 | -5
+     | 8 | 8 | eight | 3 | -3 | 0 |   
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+     | 8 | 8 | eight | 3 | -3 |   |  0
+     | 0 |   | zero  | 3 | -3 | 1 | -1
+     | 0 |   | zero  | 3 | -3 | 2 |  2
      | 0 |   | zero  | 3 | -3 | 3 | -3
-     | 0 |   | zero  | 2 |  4 | 3 | -3
-     | 0 |   | zero  | 5 | -5 | 3 | -3
-     | 0 |   | zero  | 5 | -5 | 3 | -3
-     | 0 |   | zero  | 0 |    | 3 | -3
-     | 0 |   | zero  |   |    | 3 | -3
-     | 0 |   | zero  |   |  0 | 3 | -3
-     |   |   | null  | 1 | -1 | 3 | -3
-     |   |   | null  | 2 |  2 | 3 | -3
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+     | 0 |   | zero  | 3 | -3 | 5 | -5
+     | 0 |   | zero  | 3 | -3 | 5 | -5
+     | 0 |   | zero  | 3 | -3 | 0 |   
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+     |   |   | null  | 3 | -3 | 1 | -1
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      |   |   | null  | 3 | -3 | 3 | -3
-     |   |   | null  | 2 |  4 | 3 | -3
-     |   |   | null  | 5 | -5 | 3 | -3
-     |   |   | null  | 5 | -5 | 3 | -3
-     |   |   | null  | 0 |    | 3 | -3
-     |   |   | null  |   |    | 3 | -3
-     |   |   | null  |   |  0 | 3 | -3
-     |   | 0 | zero  | 1 | -1 | 3 | -3
-     |   | 0 | zero  | 2 |  2 | 3 | -3
-     |   | 0 | zero  | 3 | -3 | 3 | -3
-     |   | 0 | zero  | 2 |  4 | 3 | -3
-     |   | 0 | zero  | 5 | -5 | 3 | -3
-     |   | 0 | zero  | 5 | -5 | 3 | -3
-     |   | 0 | zero  | 0 |    | 3 | -3
-     |   | 0 | zero  |   |    | 3 | -3
-     |   | 0 | zero  |   |  0 | 3 | -3
-     | 1 | 4 | one   | 1 | -1 | 2 |  4
-     | 1 | 4 | one   | 2 |  2 | 2 |  4
-     | 1 | 4 | one   | 3 | -3 | 2 |  4
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+     |   |   | null  | 3 | -3 | 5 | -5
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+     |   | 0 | zero  | 3 | -3 | 0 |   
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+     | 1 | 4 | one   | 2 |  4 | 1 | -1
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      | 1 | 4 | one   | 2 |  4 | 2 |  4
-     | 1 | 4 | one   | 5 | -5 | 2 |  4
-     | 1 | 4 | one   | 5 | -5 | 2 |  4
-     | 1 | 4 | one   | 0 |    | 2 |  4
-     | 1 | 4 | one   |   |    | 2 |  4
-     | 1 | 4 | one   |   |  0 | 2 |  4
-     | 2 | 3 | two   | 1 | -1 | 2 |  4
-     | 2 | 3 | two   | 2 |  2 | 2 |  4
-     | 2 | 3 | two   | 3 | -3 | 2 |  4
+     | 1 | 4 | one   | 2 |  4 | 5 | -5
+     | 1 | 4 | one   | 2 |  4 | 5 | -5
+     | 1 | 4 | one   | 2 |  4 | 0 |   
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      | 2 | 3 | two   | 2 |  4 | 2 |  4
-     | 2 | 3 | two   | 5 | -5 | 2 |  4
-     | 2 | 3 | two   | 5 | -5 | 2 |  4
-     | 2 | 3 | two   | 0 |    | 2 |  4
-     | 2 | 3 | two   |   |    | 2 |  4
-     | 2 | 3 | two   |   |  0 | 2 |  4
-     | 3 | 2 | three | 1 | -1 | 2 |  4
-     | 3 | 2 | three | 2 |  2 | 2 |  4
-     | 3 | 2 | three | 3 | -3 | 2 |  4
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+     | 2 | 3 | two   | 2 |  4 | 5 | -5
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      | 3 | 2 | three | 2 |  4 | 2 |  4
-     | 3 | 2 | three | 5 | -5 | 2 |  4
-     | 3 | 2 | three | 5 | -5 | 2 |  4
-     | 3 | 2 | three | 0 |    | 2 |  4
-     | 3 | 2 | three |   |    | 2 |  4
-     | 3 | 2 | three |   |  0 | 2 |  4
-     | 4 | 1 | four  | 1 | -1 | 2 |  4
-     | 4 | 1 | four  | 2 |  2 | 2 |  4
-     | 4 | 1 | four  | 3 | -3 | 2 |  4
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+     | 4 | 1 | four  | 2 |  4 | 1 | -1
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+     | 4 | 1 | four  | 2 |  4 | 3 | -3
      | 4 | 1 | four  | 2 |  4 | 2 |  4
-     | 4 | 1 | four  | 5 | -5 | 2 |  4
-     | 4 | 1 | four  | 5 | -5 | 2 |  4
-     | 4 | 1 | four  | 0 |    | 2 |  4
-     | 4 | 1 | four  |   |    | 2 |  4
-     | 4 | 1 | four  |   |  0 | 2 |  4
-     | 5 | 0 | five  | 1 | -1 | 2 |  4
-     | 5 | 0 | five  | 2 |  2 | 2 |  4
-     | 5 | 0 | five  | 3 | -3 | 2 |  4
+     | 4 | 1 | four  | 2 |  4 | 5 | -5
+     | 4 | 1 | four  | 2 |  4 | 5 | -5
+     | 4 | 1 | four  | 2 |  4 | 0 |   
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+     | 5 | 0 | five  | 2 |  4 | 1 | -1
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      | 5 | 0 | five  | 2 |  4 | 2 |  4
-     | 5 | 0 | five  | 5 | -5 | 2 |  4
-     | 5 | 0 | five  | 5 | -5 | 2 |  4
-     | 5 | 0 | five  | 0 |    | 2 |  4
-     | 5 | 0 | five  |   |    | 2 |  4
-     | 5 | 0 | five  |   |  0 | 2 |  4
-     | 6 | 6 | six   | 1 | -1 | 2 |  4
-     | 6 | 6 | six   | 2 |  2 | 2 |  4
-     | 6 | 6 | six   | 3 | -3 | 2 |  4
+     | 5 | 0 | five  | 2 |  4 | 5 | -5
+     | 5 | 0 | five  | 2 |  4 | 5 | -5
+     | 5 | 0 | five  | 2 |  4 | 0 |   
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+     | 6 | 6 | six   | 2 |  4 | 1 | -1
+     | 6 | 6 | six   | 2 |  4 | 2 |  2
+     | 6 | 6 | six   | 2 |  4 | 3 | -3
      | 6 | 6 | six   | 2 |  4 | 2 |  4
-     | 6 | 6 | six   | 5 | -5 | 2 |  4
-     | 6 | 6 | six   | 5 | -5 | 2 |  4
-     | 6 | 6 | six   | 0 |    | 2 |  4
-     | 6 | 6 | six   |   |    | 2 |  4
-     | 6 | 6 | six   |   |  0 | 2 |  4
-     | 7 | 7 | seven | 1 | -1 | 2 |  4
-     | 7 | 7 | seven | 2 |  2 | 2 |  4
-     | 7 | 7 | seven | 3 | -3 | 2 |  4
+     | 6 | 6 | six   | 2 |  4 | 5 | -5
+     | 6 | 6 | six   | 2 |  4 | 5 | -5
+     | 6 | 6 | six   | 2 |  4 | 0 |   
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+     | 7 | 7 | seven | 2 |  4 | 1 | -1
+     | 7 | 7 | seven | 2 |  4 | 2 |  2
+     | 7 | 7 | seven | 2 |  4 | 3 | -3
      | 7 | 7 | seven | 2 |  4 | 2 |  4
-     | 7 | 7 | seven | 5 | -5 | 2 |  4
-     | 7 | 7 | seven | 5 | -5 | 2 |  4
-     | 7 | 7 | seven | 0 |    | 2 |  4
-     | 7 | 7 | seven |   |    | 2 |  4
-     | 7 | 7 | seven |   |  0 | 2 |  4
-     | 8 | 8 | eight | 1 | -1 | 2 |  4
-     | 8 | 8 | eight | 2 |  2 | 2 |  4
-     | 8 | 8 | eight | 3 | -3 | 2 |  4
+     | 7 | 7 | seven | 2 |  4 | 5 | -5
+     | 7 | 7 | seven | 2 |  4 | 5 | -5
+     | 7 | 7 | seven | 2 |  4 | 0 |   
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+     | 8 | 8 | eight | 2 |  4 | 1 | -1
+     | 8 | 8 | eight | 2 |  4 | 2 |  2
+     | 8 | 8 | eight | 2 |  4 | 3 | -3
      | 8 | 8 | eight | 2 |  4 | 2 |  4
-     | 8 | 8 | eight | 5 | -5 | 2 |  4
-     | 8 | 8 | eight | 5 | -5 | 2 |  4
-     | 8 | 8 | eight | 0 |    | 2 |  4
-     | 8 | 8 | eight |   |    | 2 |  4
-     | 8 | 8 | eight |   |  0 | 2 |  4
-     | 0 |   | zero  | 1 | -1 | 2 |  4
-     | 0 |   | zero  | 2 |  2 | 2 |  4
-     | 0 |   | zero  | 3 | -3 | 2 |  4
+     | 8 | 8 | eight | 2 |  4 | 5 | -5
+     | 8 | 8 | eight | 2 |  4 | 5 | -5
+     | 8 | 8 | eight | 2 |  4 | 0 |   
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+     | 8 | 8 | eight | 2 |  4 |   |  0
+     | 0 |   | zero  | 2 |  4 | 1 | -1
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+     | 0 |   | zero  | 2 |  4 | 3 | -3
      | 0 |   | zero  | 2 |  4 | 2 |  4
-     | 0 |   | zero  | 5 | -5 | 2 |  4
-     | 0 |   | zero  | 5 | -5 | 2 |  4
-     | 0 |   | zero  | 0 |    | 2 |  4
-     | 0 |   | zero  |   |    | 2 |  4
-     | 0 |   | zero  |   |  0 | 2 |  4
-     |   |   | null  | 1 | -1 | 2 |  4
-     |   |   | null  | 2 |  2 | 2 |  4
-     |   |   | null  | 3 | -3 | 2 |  4
+     | 0 |   | zero  | 2 |  4 | 5 | -5
+     | 0 |   | zero  | 2 |  4 | 5 | -5
+     | 0 |   | zero  | 2 |  4 | 0 |   
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+     |   |   | null  | 2 |  4 | 1 | -1
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      |   |   | null  | 2 |  4 | 2 |  4
-     |   |   | null  | 5 | -5 | 2 |  4
-     |   |   | null  | 5 | -5 | 2 |  4
-     |   |   | null  | 0 |    | 2 |  4
-     |   |   | null  |   |    | 2 |  4
-     |   |   | null  |   |  0 | 2 |  4
-     |   | 0 | zero  | 1 | -1 | 2 |  4
-     |   | 0 | zero  | 2 |  2 | 2 |  4
-     |   | 0 | zero  | 3 | -3 | 2 |  4
+     |   |   | null  | 2 |  4 | 5 | -5
+     |   |   | null  | 2 |  4 | 5 | -5
+     |   |   | null  | 2 |  4 | 0 |   
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+     |   |   | null  | 2 |  4 |   |  0
+     |   | 0 | zero  | 2 |  4 | 1 | -1
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      |   | 0 | zero  | 2 |  4 | 2 |  4
-     |   | 0 | zero  | 5 | -5 | 2 |  4
-     |   | 0 | zero  | 5 | -5 | 2 |  4
-     |   | 0 | zero  | 0 |    | 2 |  4
-     |   | 0 | zero  |   |    | 2 |  4
-     |   | 0 | zero  |   |  0 | 2 |  4
-     | 1 | 4 | one   | 1 | -1 | 5 | -5
-     | 1 | 4 | one   | 2 |  2 | 5 | -5
-     | 1 | 4 | one   | 3 | -3 | 5 | -5
-     | 1 | 4 | one   | 2 |  4 | 5 | -5
+     |   | 0 | zero  | 2 |  4 | 5 | -5
+     |   | 0 | zero  | 2 |  4 | 5 | -5
+     |   | 0 | zero  | 2 |  4 | 0 |   
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+     |   | 0 | zero  | 2 |  4 |   |  0
+     | 1 | 4 | one   | 5 | -5 | 1 | -1
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+     | 1 | 4 | one   | 5 | -5 | 3 | -3
+     | 1 | 4 | one   | 5 | -5 | 2 |  4
      | 1 | 4 | one   | 5 | -5 | 5 | -5
      | 1 | 4 | one   | 5 | -5 | 5 | -5
-     | 1 | 4 | one   | 0 |    | 5 | -5
-     | 1 | 4 | one   |   |    | 5 | -5
-     | 1 | 4 | one   |   |  0 | 5 | -5
-     | 2 | 3 | two   | 1 | -1 | 5 | -5
-     | 2 | 3 | two   | 2 |  2 | 5 | -5
-     | 2 | 3 | two   | 3 | -3 | 5 | -5
-     | 2 | 3 | two   | 2 |  4 | 5 | -5
+     | 1 | 4 | one   | 5 | -5 | 0 |   
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      | 2 | 3 | two   | 5 | -5 | 5 | -5
      | 2 | 3 | two   | 5 | -5 | 5 | -5
-     | 2 | 3 | two   | 0 |    | 5 | -5
-     | 2 | 3 | two   |   |    | 5 | -5
-     | 2 | 3 | two   |   |  0 | 5 | -5
-     | 3 | 2 | three | 1 | -1 | 5 | -5
-     | 3 | 2 | three | 2 |  2 | 5 | -5
-     | 3 | 2 | three | 3 | -3 | 5 | -5
-     | 3 | 2 | three | 2 |  4 | 5 | -5
+     | 2 | 3 | two   | 5 | -5 | 0 |   
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+     | 3 | 2 | three | 5 | -5 | 1 | -1
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      | 3 | 2 | three | 5 | -5 | 5 | -5
      | 3 | 2 | three | 5 | -5 | 5 | -5
-     | 3 | 2 | three | 0 |    | 5 | -5
-     | 3 | 2 | three |   |    | 5 | -5
-     | 3 | 2 | three |   |  0 | 5 | -5
-     | 4 | 1 | four  | 1 | -1 | 5 | -5
-     | 4 | 1 | four  | 2 |  2 | 5 | -5
-     | 4 | 1 | four  | 3 | -3 | 5 | -5
-     | 4 | 1 | four  | 2 |  4 | 5 | -5
+     | 3 | 2 | three | 5 | -5 | 0 |   
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+     | 4 | 1 | four  | 5 | -5 | 1 | -1
+     | 4 | 1 | four  | 5 | -5 | 2 |  2
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+     | 4 | 1 | four  | 5 | -5 | 2 |  4
      | 4 | 1 | four  | 5 | -5 | 5 | -5
      | 4 | 1 | four  | 5 | -5 | 5 | -5
-     | 4 | 1 | four  | 0 |    | 5 | -5
-     | 4 | 1 | four  |   |    | 5 | -5
-     | 4 | 1 | four  |   |  0 | 5 | -5
-     | 5 | 0 | five  | 1 | -1 | 5 | -5
-     | 5 | 0 | five  | 2 |  2 | 5 | -5
-     | 5 | 0 | five  | 3 | -3 | 5 | -5
-     | 5 | 0 | five  | 2 |  4 | 5 | -5
+     | 4 | 1 | four  | 5 | -5 | 0 |   
+     | 4 | 1 | four  | 5 | -5 |   |   
+     | 4 | 1 | four  | 5 | -5 |   |  0
+     | 5 | 0 | five  | 5 | -5 | 1 | -1
+     | 5 | 0 | five  | 5 | -5 | 2 |  2
+     | 5 | 0 | five  | 5 | -5 | 3 | -3
+     | 5 | 0 | five  | 5 | -5 | 2 |  4
      | 5 | 0 | five  | 5 | -5 | 5 | -5
      | 5 | 0 | five  | 5 | -5 | 5 | -5
-     | 5 | 0 | five  | 0 |    | 5 | -5
-     | 5 | 0 | five  |   |    | 5 | -5
-     | 5 | 0 | five  |   |  0 | 5 | -5
-     | 6 | 6 | six   | 1 | -1 | 5 | -5
-     | 6 | 6 | six   | 2 |  2 | 5 | -5
-     | 6 | 6 | six   | 3 | -3 | 5 | -5
-     | 6 | 6 | six   | 2 |  4 | 5 | -5
+     | 5 | 0 | five  | 5 | -5 | 0 |   
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+     | 5 | 0 | five  | 5 | -5 |   |  0
+     | 6 | 6 | six   | 5 | -5 | 1 | -1
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      | 6 | 6 | six   | 5 | -5 | 5 | -5
      | 6 | 6 | six   | 5 | -5 | 5 | -5
-     | 6 | 6 | six   | 0 |    | 5 | -5
-     | 6 | 6 | six   |   |    | 5 | -5
-     | 6 | 6 | six   |   |  0 | 5 | -5
-     | 7 | 7 | seven | 1 | -1 | 5 | -5
-     | 7 | 7 | seven | 2 |  2 | 5 | -5
-     | 7 | 7 | seven | 3 | -3 | 5 | -5
-     | 7 | 7 | seven | 2 |  4 | 5 | -5
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+     | 6 | 6 | six   | 5 | -5 |   |  0
+     | 7 | 7 | seven | 5 | -5 | 1 | -1
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      | 7 | 7 | seven | 5 | -5 | 5 | -5
      | 7 | 7 | seven | 5 | -5 | 5 | -5
-     | 7 | 7 | seven | 0 |    | 5 | -5
-     | 7 | 7 | seven |   |    | 5 | -5
-     | 7 | 7 | seven |   |  0 | 5 | -5
-     | 8 | 8 | eight | 1 | -1 | 5 | -5
-     | 8 | 8 | eight | 2 |  2 | 5 | -5
-     | 8 | 8 | eight | 3 | -3 | 5 | -5
-     | 8 | 8 | eight | 2 |  4 | 5 | -5
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+     | 8 | 8 | eight | 5 | -5 | 1 | -1
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      | 8 | 8 | eight | 5 | -5 | 5 | -5
      | 8 | 8 | eight | 5 | -5 | 5 | -5
-     | 8 | 8 | eight | 0 |    | 5 | -5
-     | 8 | 8 | eight |   |    | 5 | -5
-     | 8 | 8 | eight |   |  0 | 5 | -5
-     | 0 |   | zero  | 1 | -1 | 5 | -5
-     | 0 |   | zero  | 2 |  2 | 5 | -5
-     | 0 |   | zero  | 3 | -3 | 5 | -5
-     | 0 |   | zero  | 2 |  4 | 5 | -5
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+     | 0 |   | zero  | 5 | -5 | 1 | -1
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      | 0 |   | zero  | 5 | -5 | 5 | -5
      | 0 |   | zero  | 5 | -5 | 5 | -5
-     | 0 |   | zero  | 0 |    | 5 | -5
-     | 0 |   | zero  |   |    | 5 | -5
-     | 0 |   | zero  |   |  0 | 5 | -5
-     |   |   | null  | 1 | -1 | 5 | -5
-     |   |   | null  | 2 |  2 | 5 | -5
-     |   |   | null  | 3 | -3 | 5 | -5
-     |   |   | null  | 2 |  4 | 5 | -5
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      |   |   | null  | 5 | -5 | 5 | -5
      |   |   | null  | 5 | -5 | 5 | -5
-     |   |   | null  | 0 |    | 5 | -5
-     |   |   | null  |   |    | 5 | -5
-     |   |   | null  |   |  0 | 5 | -5
-     |   | 0 | zero  | 1 | -1 | 5 | -5
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-     |   | 0 | zero  | 2 |  4 | 5 | -5
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      |   | 0 | zero  | 5 | -5 | 5 | -5
      |   | 0 | zero  | 5 | -5 | 5 | -5
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-     |   | 0 | zero  |   |  0 | 5 | -5
-     | 1 | 4 | one   | 1 | -1 | 5 | -5
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-     | 1 | 4 | one   | 2 |  4 | 5 | -5
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      | 1 | 4 | one   | 5 | -5 | 5 | -5
      | 1 | 4 | one   | 5 | -5 | 5 | -5
-     | 1 | 4 | one   | 0 |    | 5 | -5
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-     | 2 | 3 | two   | 2 |  4 | 5 | -5
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      | 2 | 3 | two   | 5 | -5 | 5 | -5
      | 2 | 3 | two   | 5 | -5 | 5 | -5
-     | 2 | 3 | two   | 0 |    | 5 | -5
-     | 2 | 3 | two   |   |    | 5 | -5
-     | 2 | 3 | two   |   |  0 | 5 | -5
-     | 3 | 2 | three | 1 | -1 | 5 | -5
-     | 3 | 2 | three | 2 |  2 | 5 | -5
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-     | 3 | 2 | three | 2 |  4 | 5 | -5
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      | 3 | 2 | three | 5 | -5 | 5 | -5
      | 3 | 2 | three | 5 | -5 | 5 | -5
-     | 3 | 2 | three | 0 |    | 5 | -5
-     | 3 | 2 | three |   |    | 5 | -5
-     | 3 | 2 | three |   |  0 | 5 | -5
-     | 4 | 1 | four  | 1 | -1 | 5 | -5
-     | 4 | 1 | four  | 2 |  2 | 5 | -5
-     | 4 | 1 | four  | 3 | -3 | 5 | -5
-     | 4 | 1 | four  | 2 |  4 | 5 | -5
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      | 4 | 1 | four  | 5 | -5 | 5 | -5
      | 4 | 1 | four  | 5 | -5 | 5 | -5
-     | 4 | 1 | four  | 0 |    | 5 | -5
-     | 4 | 1 | four  |   |    | 5 | -5
-     | 4 | 1 | four  |   |  0 | 5 | -5
-     | 5 | 0 | five  | 1 | -1 | 5 | -5
-     | 5 | 0 | five  | 2 |  2 | 5 | -5
-     | 5 | 0 | five  | 3 | -3 | 5 | -5
-     | 5 | 0 | five  | 2 |  4 | 5 | -5
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      | 5 | 0 | five  | 5 | -5 | 5 | -5
      | 5 | 0 | five  | 5 | -5 | 5 | -5
-     | 5 | 0 | five  | 0 |    | 5 | -5
-     | 5 | 0 | five  |   |    | 5 | -5
-     | 5 | 0 | five  |   |  0 | 5 | -5
-     | 6 | 6 | six   | 1 | -1 | 5 | -5
-     | 6 | 6 | six   | 2 |  2 | 5 | -5
-     | 6 | 6 | six   | 3 | -3 | 5 | -5
-     | 6 | 6 | six   | 2 |  4 | 5 | -5
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      | 6 | 6 | six   | 5 | -5 | 5 | -5
      | 6 | 6 | six   | 5 | -5 | 5 | -5
-     | 6 | 6 | six   | 0 |    | 5 | -5
-     | 6 | 6 | six   |   |    | 5 | -5
-     | 6 | 6 | six   |   |  0 | 5 | -5
-     | 7 | 7 | seven | 1 | -1 | 5 | -5
-     | 7 | 7 | seven | 2 |  2 | 5 | -5
-     | 7 | 7 | seven | 3 | -3 | 5 | -5
-     | 7 | 7 | seven | 2 |  4 | 5 | -5
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      | 7 | 7 | seven | 5 | -5 | 5 | -5
      | 7 | 7 | seven | 5 | -5 | 5 | -5
-     | 7 | 7 | seven | 0 |    | 5 | -5
-     | 7 | 7 | seven |   |    | 5 | -5
-     | 7 | 7 | seven |   |  0 | 5 | -5
-     | 8 | 8 | eight | 1 | -1 | 5 | -5
-     | 8 | 8 | eight | 2 |  2 | 5 | -5
-     | 8 | 8 | eight | 3 | -3 | 5 | -5
-     | 8 | 8 | eight | 2 |  4 | 5 | -5
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+     | 8 | 8 | eight | 5 | -5 | 1 | -1
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      | 8 | 8 | eight | 5 | -5 | 5 | -5
      | 8 | 8 | eight | 5 | -5 | 5 | -5
-     | 8 | 8 | eight | 0 |    | 5 | -5
-     | 8 | 8 | eight |   |    | 5 | -5
-     | 8 | 8 | eight |   |  0 | 5 | -5
-     | 0 |   | zero  | 1 | -1 | 5 | -5
-     | 0 |   | zero  | 2 |  2 | 5 | -5
-     | 0 |   | zero  | 3 | -3 | 5 | -5
-     | 0 |   | zero  | 2 |  4 | 5 | -5
+     | 8 | 8 | eight | 5 | -5 | 0 |   
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+     | 0 |   | zero  | 5 | -5 | 1 | -1
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+     | 0 |   | zero  | 5 | -5 | 2 |  4
      | 0 |   | zero  | 5 | -5 | 5 | -5
      | 0 |   | zero  | 5 | -5 | 5 | -5
-     | 0 |   | zero  | 0 |    | 5 | -5
-     | 0 |   | zero  |   |    | 5 | -5
-     | 0 |   | zero  |   |  0 | 5 | -5
-     |   |   | null  | 1 | -1 | 5 | -5
-     |   |   | null  | 2 |  2 | 5 | -5
-     |   |   | null  | 3 | -3 | 5 | -5
-     |   |   | null  | 2 |  4 | 5 | -5
+     | 0 |   | zero  | 5 | -5 | 0 |   
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      |   |   | null  | 5 | -5 | 5 | -5
      |   |   | null  | 5 | -5 | 5 | -5
-     |   |   | null  | 0 |    | 5 | -5
-     |   |   | null  |   |    | 5 | -5
-     |   |   | null  |   |  0 | 5 | -5
-     |   | 0 | zero  | 1 | -1 | 5 | -5
-     |   | 0 | zero  | 2 |  2 | 5 | -5
-     |   | 0 | zero  | 3 | -3 | 5 | -5
-     |   | 0 | zero  | 2 |  4 | 5 | -5
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      |   | 0 | zero  | 5 | -5 | 5 | -5
      |   | 0 | zero  | 5 | -5 | 5 | -5
-     |   | 0 | zero  | 0 |    | 5 | -5
-     |   | 0 | zero  |   |    | 5 | -5
-     |   | 0 | zero  |   |  0 | 5 | -5
-     | 1 | 4 | one   | 1 | -1 | 0 |   
-     | 1 | 4 | one   | 2 |  2 | 0 |   
-     | 1 | 4 | one   | 3 | -3 | 0 |   
-     | 1 | 4 | one   | 2 |  4 | 0 |   
-     | 1 | 4 | one   | 5 | -5 | 0 |   
-     | 1 | 4 | one   | 5 | -5 | 0 |   
-     | 1 | 4 | one   | 0 |    | 0 |   
-     | 1 | 4 | one   |   |    | 0 |   
-     | 1 | 4 | one   |   |  0 | 0 |   
-     | 2 | 3 | two   | 1 | -1 | 0 |   
-     | 2 | 3 | two   | 2 |  2 | 0 |   
-     | 2 | 3 | two   | 3 | -3 | 0 |   
-     | 2 | 3 | two   | 2 |  4 | 0 |   
-     | 2 | 3 | two   | 5 | -5 | 0 |   
-     | 2 | 3 | two   | 5 | -5 | 0 |   
+     |   | 0 | zero  | 5 | -5 | 0 |   
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+     | 1 | 4 | one   | 0 |    | 1 | -1
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      | 2 | 3 | two   | 0 |    | 0 |   
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-     | 2 | 3 | two   |   |  0 | 0 |   
-     | 3 | 2 | three | 1 | -1 | 0 |   
-     | 3 | 2 | three | 2 |  2 | 0 |   
-     | 3 | 2 | three | 3 | -3 | 0 |   
-     | 3 | 2 | three | 2 |  4 | 0 |   
-     | 3 | 2 | three | 5 | -5 | 0 |   
-     | 3 | 2 | three | 5 | -5 | 0 |   
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      | 3 | 2 | three | 0 |    | 0 |   
-     | 3 | 2 | three |   |    | 0 |   
-     | 3 | 2 | three |   |  0 | 0 |   
-     | 4 | 1 | four  | 1 | -1 | 0 |   
-     | 4 | 1 | four  | 2 |  2 | 0 |   
-     | 4 | 1 | four  | 3 | -3 | 0 |   
-     | 4 | 1 | four  | 2 |  4 | 0 |   
-     | 4 | 1 | four  | 5 | -5 | 0 |   
-     | 4 | 1 | four  | 5 | -5 | 0 |   
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      | 4 | 1 | four  | 0 |    | 0 |   
-     | 4 | 1 | four  |   |    | 0 |   
-     | 4 | 1 | four  |   |  0 | 0 |   
-     | 5 | 0 | five  | 1 | -1 | 0 |   
-     | 5 | 0 | five  | 2 |  2 | 0 |   
-     | 5 | 0 | five  | 3 | -3 | 0 |   
-     | 5 | 0 | five  | 2 |  4 | 0 |   
-     | 5 | 0 | five  | 5 | -5 | 0 |   
-     | 5 | 0 | five  | 5 | -5 | 0 |   
+     | 4 | 1 | four  | 0 |    |   |   
+     | 4 | 1 | four  | 0 |    |   |  0
+     | 5 | 0 | five  | 0 |    | 1 | -1
+     | 5 | 0 | five  | 0 |    | 2 |  2
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+     | 5 | 0 | five  | 0 |    | 2 |  4
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+     | 5 | 0 | five  | 0 |    | 5 | -5
      | 5 | 0 | five  | 0 |    | 0 |   
-     | 5 | 0 | five  |   |    | 0 |   
-     | 5 | 0 | five  |   |  0 | 0 |   
-     | 6 | 6 | six   | 1 | -1 | 0 |   
-     | 6 | 6 | six   | 2 |  2 | 0 |   
-     | 6 | 6 | six   | 3 | -3 | 0 |   
-     | 6 | 6 | six   | 2 |  4 | 0 |   
-     | 6 | 6 | six   | 5 | -5 | 0 |   
-     | 6 | 6 | six   | 5 | -5 | 0 |   
+     | 5 | 0 | five  | 0 |    |   |   
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+     | 6 | 6 | six   | 0 |    | 1 | -1
+     | 6 | 6 | six   | 0 |    | 2 |  2
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+     | 6 | 6 | six   | 0 |    | 5 | -5
      | 6 | 6 | six   | 0 |    | 0 |   
-     | 6 | 6 | six   |   |    | 0 |   
-     | 6 | 6 | six   |   |  0 | 0 |   
-     | 7 | 7 | seven | 1 | -1 | 0 |   
-     | 7 | 7 | seven | 2 |  2 | 0 |   
-     | 7 | 7 | seven | 3 | -3 | 0 |   
-     | 7 | 7 | seven | 2 |  4 | 0 |   
-     | 7 | 7 | seven | 5 | -5 | 0 |   
-     | 7 | 7 | seven | 5 | -5 | 0 |   
+     | 6 | 6 | six   | 0 |    |   |   
+     | 6 | 6 | six   | 0 |    |   |  0
+     | 7 | 7 | seven | 0 |    | 1 | -1
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      | 7 | 7 | seven | 0 |    | 0 |   
-     | 7 | 7 | seven |   |    | 0 |   
-     | 7 | 7 | seven |   |  0 | 0 |   
-     | 8 | 8 | eight | 1 | -1 | 0 |   
-     | 8 | 8 | eight | 2 |  2 | 0 |   
-     | 8 | 8 | eight | 3 | -3 | 0 |   
-     | 8 | 8 | eight | 2 |  4 | 0 |   
-     | 8 | 8 | eight | 5 | -5 | 0 |   
-     | 8 | 8 | eight | 5 | -5 | 0 |   
+     | 7 | 7 | seven | 0 |    |   |   
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+     | 8 | 8 | eight | 0 |    | 2 |  2
+     | 8 | 8 | eight | 0 |    | 3 | -3
+     | 8 | 8 | eight | 0 |    | 2 |  4
+     | 8 | 8 | eight | 0 |    | 5 | -5
+     | 8 | 8 | eight | 0 |    | 5 | -5
      | 8 | 8 | eight | 0 |    | 0 |   
-     | 8 | 8 | eight |   |    | 0 |   
-     | 8 | 8 | eight |   |  0 | 0 |   
-     | 0 |   | zero  | 1 | -1 | 0 |   
-     | 0 |   | zero  | 2 |  2 | 0 |   
-     | 0 |   | zero  | 3 | -3 | 0 |   
-     | 0 |   | zero  | 2 |  4 | 0 |   
-     | 0 |   | zero  | 5 | -5 | 0 |   
-     | 0 |   | zero  | 5 | -5 | 0 |   
+     | 8 | 8 | eight | 0 |    |   |   
+     | 8 | 8 | eight | 0 |    |   |  0
+     | 0 |   | zero  | 0 |    | 1 | -1
+     | 0 |   | zero  | 0 |    | 2 |  2
+     | 0 |   | zero  | 0 |    | 3 | -3
+     | 0 |   | zero  | 0 |    | 2 |  4
+     | 0 |   | zero  | 0 |    | 5 | -5
+     | 0 |   | zero  | 0 |    | 5 | -5
      | 0 |   | zero  | 0 |    | 0 |   
-     | 0 |   | zero  |   |    | 0 |   
-     | 0 |   | zero  |   |  0 | 0 |   
-     |   |   | null  | 1 | -1 | 0 |   
-     |   |   | null  | 2 |  2 | 0 |   
-     |   |   | null  | 3 | -3 | 0 |   
-     |   |   | null  | 2 |  4 | 0 |   
-     |   |   | null  | 5 | -5 | 0 |   
-     |   |   | null  | 5 | -5 | 0 |   
+     | 0 |   | zero  | 0 |    |   |   
+     | 0 |   | zero  | 0 |    |   |  0
+     |   |   | null  | 0 |    | 1 | -1
+     |   |   | null  | 0 |    | 2 |  2
+     |   |   | null  | 0 |    | 3 | -3
+     |   |   | null  | 0 |    | 2 |  4
+     |   |   | null  | 0 |    | 5 | -5
+     |   |   | null  | 0 |    | 5 | -5
      |   |   | null  | 0 |    | 0 |   
-     |   |   | null  |   |    | 0 |   
-     |   |   | null  |   |  0 | 0 |   
-     |   | 0 | zero  | 1 | -1 | 0 |   
-     |   | 0 | zero  | 2 |  2 | 0 |   
-     |   | 0 | zero  | 3 | -3 | 0 |   
-     |   | 0 | zero  | 2 |  4 | 0 |   
-     |   | 0 | zero  | 5 | -5 | 0 |   
-     |   | 0 | zero  | 5 | -5 | 0 |   
+     |   |   | null  | 0 |    |   |   
+     |   |   | null  | 0 |    |   |  0
+     |   | 0 | zero  | 0 |    | 1 | -1
+     |   | 0 | zero  | 0 |    | 2 |  2
+     |   | 0 | zero  | 0 |    | 3 | -3
+     |   | 0 | zero  | 0 |    | 2 |  4
+     |   | 0 | zero  | 0 |    | 5 | -5
+     |   | 0 | zero  | 0 |    | 5 | -5
      |   | 0 | zero  | 0 |    | 0 |   
-     |   | 0 | zero  |   |    | 0 |   
-     |   | 0 | zero  |   |  0 | 0 |   
-     | 1 | 4 | one   | 1 | -1 |   |   
-     | 1 | 4 | one   | 2 |  2 |   |   
-     | 1 | 4 | one   | 3 | -3 |   |   
-     | 1 | 4 | one   | 2 |  4 |   |   
-     | 1 | 4 | one   | 5 | -5 |   |   
-     | 1 | 4 | one   | 5 | -5 |   |   
-     | 1 | 4 | one   | 0 |    |   |   
+     |   | 0 | zero  | 0 |    |   |   
+     |   | 0 | zero  | 0 |    |   |  0
+     | 1 | 4 | one   |   |    | 1 | -1
+     | 1 | 4 | one   |   |    | 2 |  2
+     | 1 | 4 | one   |   |    | 3 | -3
+     | 1 | 4 | one   |   |    | 2 |  4
+     | 1 | 4 | one   |   |    | 5 | -5
+     | 1 | 4 | one   |   |    | 5 | -5
+     | 1 | 4 | one   |   |    | 0 |   
      | 1 | 4 | one   |   |    |   |   
-     | 1 | 4 | one   |   |  0 |   |   
-     | 2 | 3 | two   | 1 | -1 |   |   
-     | 2 | 3 | two   | 2 |  2 |   |   
-     | 2 | 3 | two   | 3 | -3 |   |   
-     | 2 | 3 | two   | 2 |  4 |   |   
-     | 2 | 3 | two   | 5 | -5 |   |   
-     | 2 | 3 | two   | 5 | -5 |   |   
-     | 2 | 3 | two   | 0 |    |   |   
+     | 1 | 4 | one   |   |    |   |  0
+     | 2 | 3 | two   |   |    | 1 | -1
+     | 2 | 3 | two   |   |    | 2 |  2
+     | 2 | 3 | two   |   |    | 3 | -3
+     | 2 | 3 | two   |   |    | 2 |  4
+     | 2 | 3 | two   |   |    | 5 | -5
+     | 2 | 3 | two   |   |    | 5 | -5
+     | 2 | 3 | two   |   |    | 0 |   
      | 2 | 3 | two   |   |    |   |   
-     | 2 | 3 | two   |   |  0 |   |   
-     | 3 | 2 | three | 1 | -1 |   |   
-     | 3 | 2 | three | 2 |  2 |   |   
-     | 3 | 2 | three | 3 | -3 |   |   
-     | 3 | 2 | three | 2 |  4 |   |   
-     | 3 | 2 | three | 5 | -5 |   |   
-     | 3 | 2 | three | 5 | -5 |   |   
-     | 3 | 2 | three | 0 |    |   |   
+     | 2 | 3 | two   |   |    |   |  0
+     | 3 | 2 | three |   |    | 1 | -1
+     | 3 | 2 | three |   |    | 2 |  2
+     | 3 | 2 | three |   |    | 3 | -3
+     | 3 | 2 | three |   |    | 2 |  4
+     | 3 | 2 | three |   |    | 5 | -5
+     | 3 | 2 | three |   |    | 5 | -5
+     | 3 | 2 | three |   |    | 0 |   
      | 3 | 2 | three |   |    |   |   
-     | 3 | 2 | three |   |  0 |   |   
-     | 4 | 1 | four  | 1 | -1 |   |   
-     | 4 | 1 | four  | 2 |  2 |   |   
-     | 4 | 1 | four  | 3 | -3 |   |   
-     | 4 | 1 | four  | 2 |  4 |   |   
-     | 4 | 1 | four  | 5 | -5 |   |   
-     | 4 | 1 | four  | 5 | -5 |   |   
-     | 4 | 1 | four  | 0 |    |   |   
+     | 3 | 2 | three |   |    |   |  0
+     | 4 | 1 | four  |   |    | 1 | -1
+     | 4 | 1 | four  |   |    | 2 |  2
+     | 4 | 1 | four  |   |    | 3 | -3
+     | 4 | 1 | four  |   |    | 2 |  4
+     | 4 | 1 | four  |   |    | 5 | -5
+     | 4 | 1 | four  |   |    | 5 | -5
+     | 4 | 1 | four  |   |    | 0 |   
      | 4 | 1 | four  |   |    |   |   
-     | 4 | 1 | four  |   |  0 |   |   
-     | 5 | 0 | five  | 1 | -1 |   |   
-     | 5 | 0 | five  | 2 |  2 |   |   
-     | 5 | 0 | five  | 3 | -3 |   |   
-     | 5 | 0 | five  | 2 |  4 |   |   
-     | 5 | 0 | five  | 5 | -5 |   |   
-     | 5 | 0 | five  | 5 | -5 |   |   
-     | 5 | 0 | five  | 0 |    |   |   
+     | 4 | 1 | four  |   |    |   |  0
+     | 5 | 0 | five  |   |    | 1 | -1
+     | 5 | 0 | five  |   |    | 2 |  2
+     | 5 | 0 | five  |   |    | 3 | -3
+     | 5 | 0 | five  |   |    | 2 |  4
+     | 5 | 0 | five  |   |    | 5 | -5
+     | 5 | 0 | five  |   |    | 5 | -5
+     | 5 | 0 | five  |   |    | 0 |   
      | 5 | 0 | five  |   |    |   |   
-     | 5 | 0 | five  |   |  0 |   |   
-     | 6 | 6 | six   | 1 | -1 |   |   
-     | 6 | 6 | six   | 2 |  2 |   |   
-     | 6 | 6 | six   | 3 | -3 |   |   
-     | 6 | 6 | six   | 2 |  4 |   |   
-     | 6 | 6 | six   | 5 | -5 |   |   
-     | 6 | 6 | six   | 5 | -5 |   |   
-     | 6 | 6 | six   | 0 |    |   |   
-     | 6 | 6 | six   |   |    |   |   
-     | 6 | 6 | six   |   |  0 |   |   
-     | 7 | 7 | seven | 1 | -1 |   |   
-     | 7 | 7 | seven | 2 |  2 |   |   
-     | 7 | 7 | seven | 3 | -3 |   |   
-     | 7 | 7 | seven | 2 |  4 |   |   
-     | 7 | 7 | seven | 5 | -5 |   |   
-     | 7 | 7 | seven | 5 | -5 |   |   
-     | 7 | 7 | seven | 0 |    |   |   
+     | 5 | 0 | five  |   |    |   |  0
+     | 6 | 6 | six   |   |    | 1 | -1
+     | 6 | 6 | six   |   |    | 2 |  2
+     | 6 | 6 | six   |   |    | 3 | -3
+     | 6 | 6 | six   |   |    | 2 |  4
+     | 6 | 6 | six   |   |    | 5 | -5
+     | 6 | 6 | six   |   |    | 5 | -5
+     | 6 | 6 | six   |   |    | 0 |   
+     | 6 | 6 | six   |   |    |   |   
+     | 6 | 6 | six   |   |    |   |  0
+     | 7 | 7 | seven |   |    | 1 | -1
+     | 7 | 7 | seven |   |    | 2 |  2
+     | 7 | 7 | seven |   |    | 3 | -3
+     | 7 | 7 | seven |   |    | 2 |  4
+     | 7 | 7 | seven |   |    | 5 | -5
+     | 7 | 7 | seven |   |    | 5 | -5
+     | 7 | 7 | seven |   |    | 0 |   
      | 7 | 7 | seven |   |    |   |   
-     | 7 | 7 | seven |   |  0 |   |   
-     | 8 | 8 | eight | 1 | -1 |   |   
-     | 8 | 8 | eight | 2 |  2 |   |   
-     | 8 | 8 | eight | 3 | -3 |   |   
-     | 8 | 8 | eight | 2 |  4 |   |   
-     | 8 | 8 | eight | 5 | -5 |   |   
-     | 8 | 8 | eight | 5 | -5 |   |   
-     | 8 | 8 | eight | 0 |    |   |   
+     | 7 | 7 | seven |   |    |   |  0
+     | 8 | 8 | eight |   |    | 1 | -1
+     | 8 | 8 | eight |   |    | 2 |  2
+     | 8 | 8 | eight |   |    | 3 | -3
+     | 8 | 8 | eight |   |    | 2 |  4
+     | 8 | 8 | eight |   |    | 5 | -5
+     | 8 | 8 | eight |   |    | 5 | -5
+     | 8 | 8 | eight |   |    | 0 |   
      | 8 | 8 | eight |   |    |   |   
-     | 8 | 8 | eight |   |  0 |   |   
-     | 0 |   | zero  | 1 | -1 |   |   
-     | 0 |   | zero  | 2 |  2 |   |   
-     | 0 |   | zero  | 3 | -3 |   |   
-     | 0 |   | zero  | 2 |  4 |   |   
-     | 0 |   | zero  | 5 | -5 |   |   
-     | 0 |   | zero  | 5 | -5 |   |   
-     | 0 |   | zero  | 0 |    |   |   
+     | 8 | 8 | eight |   |    |   |  0
+     | 0 |   | zero  |   |    | 1 | -1
+     | 0 |   | zero  |   |    | 2 |  2
+     | 0 |   | zero  |   |    | 3 | -3
+     | 0 |   | zero  |   |    | 2 |  4
+     | 0 |   | zero  |   |    | 5 | -5
+     | 0 |   | zero  |   |    | 5 | -5
+     | 0 |   | zero  |   |    | 0 |   
      | 0 |   | zero  |   |    |   |   
-     | 0 |   | zero  |   |  0 |   |   
-     |   |   | null  | 1 | -1 |   |   
-     |   |   | null  | 2 |  2 |   |   
-     |   |   | null  | 3 | -3 |   |   
-     |   |   | null  | 2 |  4 |   |   
-     |   |   | null  | 5 | -5 |   |   
-     |   |   | null  | 5 | -5 |   |   
-     |   |   | null  | 0 |    |   |   
+     | 0 |   | zero  |   |    |   |  0
+     |   |   | null  |   |    | 1 | -1
+     |   |   | null  |   |    | 2 |  2
+     |   |   | null  |   |    | 3 | -3
+     |   |   | null  |   |    | 2 |  4
+     |   |   | null  |   |    | 5 | -5
+     |   |   | null  |   |    | 5 | -5
+     |   |   | null  |   |    | 0 |   
      |   |   | null  |   |    |   |   
-     |   |   | null  |   |  0 |   |   
-     |   | 0 | zero  | 1 | -1 |   |   
-     |   | 0 | zero  | 2 |  2 |   |   
-     |   | 0 | zero  | 3 | -3 |   |   
-     |   | 0 | zero  | 2 |  4 |   |   
-     |   | 0 | zero  | 5 | -5 |   |   
-     |   | 0 | zero  | 5 | -5 |   |   
-     |   | 0 | zero  | 0 |    |   |   
+     |   |   | null  |   |    |   |  0
+     |   | 0 | zero  |   |    | 1 | -1
+     |   | 0 | zero  |   |    | 2 |  2
+     |   | 0 | zero  |   |    | 3 | -3
+     |   | 0 | zero  |   |    | 2 |  4
+     |   | 0 | zero  |   |    | 5 | -5
+     |   | 0 | zero  |   |    | 5 | -5
+     |   | 0 | zero  |   |    | 0 |   
      |   | 0 | zero  |   |    |   |   
-     |   | 0 | zero  |   |  0 |   |   
-     | 1 | 4 | one   | 1 | -1 |   |  0
-     | 1 | 4 | one   | 2 |  2 |   |  0
-     | 1 | 4 | one   | 3 | -3 |   |  0
-     | 1 | 4 | one   | 2 |  4 |   |  0
-     | 1 | 4 | one   | 5 | -5 |   |  0
-     | 1 | 4 | one   | 5 | -5 |   |  0
-     | 1 | 4 | one   | 0 |    |   |  0
-     | 1 | 4 | one   |   |    |   |  0
+     |   | 0 | zero  |   |    |   |  0
+     | 1 | 4 | one   |   |  0 | 1 | -1
+     | 1 | 4 | one   |   |  0 | 2 |  2
+     | 1 | 4 | one   |   |  0 | 3 | -3
+     | 1 | 4 | one   |   |  0 | 2 |  4
+     | 1 | 4 | one   |   |  0 | 5 | -5
+     | 1 | 4 | one   |   |  0 | 5 | -5
+     | 1 | 4 | one   |   |  0 | 0 |   
+     | 1 | 4 | one   |   |  0 |   |   
      | 1 | 4 | one   |   |  0 |   |  0
-     | 2 | 3 | two   | 1 | -1 |   |  0
-     | 2 | 3 | two   | 2 |  2 |   |  0
-     | 2 | 3 | two   | 3 | -3 |   |  0
-     | 2 | 3 | two   | 2 |  4 |   |  0
-     | 2 | 3 | two   | 5 | -5 |   |  0
-     | 2 | 3 | two   | 5 | -5 |   |  0
-     | 2 | 3 | two   | 0 |    |   |  0
-     | 2 | 3 | two   |   |    |   |  0
+     | 2 | 3 | two   |   |  0 | 1 | -1
+     | 2 | 3 | two   |   |  0 | 2 |  2
+     | 2 | 3 | two   |   |  0 | 3 | -3
+     | 2 | 3 | two   |   |  0 | 2 |  4
+     | 2 | 3 | two   |   |  0 | 5 | -5
+     | 2 | 3 | two   |   |  0 | 5 | -5
+     | 2 | 3 | two   |   |  0 | 0 |   
+     | 2 | 3 | two   |   |  0 |   |   
      | 2 | 3 | two   |   |  0 |   |  0
-     | 3 | 2 | three | 1 | -1 |   |  0
-     | 3 | 2 | three | 2 |  2 |   |  0
-     | 3 | 2 | three | 3 | -3 |   |  0
-     | 3 | 2 | three | 2 |  4 |   |  0
-     | 3 | 2 | three | 5 | -5 |   |  0
-     | 3 | 2 | three | 5 | -5 |   |  0
-     | 3 | 2 | three | 0 |    |   |  0
-     | 3 | 2 | three |   |    |   |  0
+     | 3 | 2 | three |   |  0 | 1 | -1
+     | 3 | 2 | three |   |  0 | 2 |  2
+     | 3 | 2 | three |   |  0 | 3 | -3
+     | 3 | 2 | three |   |  0 | 2 |  4
+     | 3 | 2 | three |   |  0 | 5 | -5
+     | 3 | 2 | three |   |  0 | 5 | -5
+     | 3 | 2 | three |   |  0 | 0 |   
+     | 3 | 2 | three |   |  0 |   |   
      | 3 | 2 | three |   |  0 |   |  0
-     | 4 | 1 | four  | 1 | -1 |   |  0
-     | 4 | 1 | four  | 2 |  2 |   |  0
-     | 4 | 1 | four  | 3 | -3 |   |  0
-     | 4 | 1 | four  | 2 |  4 |   |  0
-     | 4 | 1 | four  | 5 | -5 |   |  0
-     | 4 | 1 | four  | 5 | -5 |   |  0
-     | 4 | 1 | four  | 0 |    |   |  0
-     | 4 | 1 | four  |   |    |   |  0
+     | 4 | 1 | four  |   |  0 | 1 | -1
+     | 4 | 1 | four  |   |  0 | 2 |  2
+     | 4 | 1 | four  |   |  0 | 3 | -3
+     | 4 | 1 | four  |   |  0 | 2 |  4
+     | 4 | 1 | four  |   |  0 | 5 | -5
+     | 4 | 1 | four  |   |  0 | 5 | -5
+     | 4 | 1 | four  |   |  0 | 0 |   
+     | 4 | 1 | four  |   |  0 |   |   
      | 4 | 1 | four  |   |  0 |   |  0
-     | 5 | 0 | five  | 1 | -1 |   |  0
-     | 5 | 0 | five  | 2 |  2 |   |  0
-     | 5 | 0 | five  | 3 | -3 |   |  0
-     | 5 | 0 | five  | 2 |  4 |   |  0
-     | 5 | 0 | five  | 5 | -5 |   |  0
-     | 5 | 0 | five  | 5 | -5 |   |  0
-     | 5 | 0 | five  | 0 |    |   |  0
-     | 5 | 0 | five  |   |    |   |  0
+     | 5 | 0 | five  |   |  0 | 1 | -1
+     | 5 | 0 | five  |   |  0 | 2 |  2
+     | 5 | 0 | five  |   |  0 | 3 | -3
+     | 5 | 0 | five  |   |  0 | 2 |  4
+     | 5 | 0 | five  |   |  0 | 5 | -5
+     | 5 | 0 | five  |   |  0 | 5 | -5
+     | 5 | 0 | five  |   |  0 | 0 |   
+     | 5 | 0 | five  |   |  0 |   |   
      | 5 | 0 | five  |   |  0 |   |  0
-     | 6 | 6 | six   | 1 | -1 |   |  0
-     | 6 | 6 | six   | 2 |  2 |   |  0
-     | 6 | 6 | six   | 3 | -3 |   |  0
-     | 6 | 6 | six   | 2 |  4 |   |  0
-     | 6 | 6 | six   | 5 | -5 |   |  0
-     | 6 | 6 | six   | 5 | -5 |   |  0
-     | 6 | 6 | six   | 0 |    |   |  0
-     | 6 | 6 | six   |   |    |   |  0
+     | 6 | 6 | six   |   |  0 | 1 | -1
+     | 6 | 6 | six   |   |  0 | 2 |  2
+     | 6 | 6 | six   |   |  0 | 3 | -3
+     | 6 | 6 | six   |   |  0 | 2 |  4
+     | 6 | 6 | six   |   |  0 | 5 | -5
+     | 6 | 6 | six   |   |  0 | 5 | -5
+     | 6 | 6 | six   |   |  0 | 0 |   
+     | 6 | 6 | six   |   |  0 |   |   
      | 6 | 6 | six   |   |  0 |   |  0
-     | 7 | 7 | seven | 1 | -1 |   |  0
-     | 7 | 7 | seven | 2 |  2 |   |  0
-     | 7 | 7 | seven | 3 | -3 |   |  0
-     | 7 | 7 | seven | 2 |  4 |   |  0
-     | 7 | 7 | seven | 5 | -5 |   |  0
-     | 7 | 7 | seven | 5 | -5 |   |  0
-     | 7 | 7 | seven | 0 |    |   |  0
-     | 7 | 7 | seven |   |    |   |  0
+     | 7 | 7 | seven |   |  0 | 1 | -1
+     | 7 | 7 | seven |   |  0 | 2 |  2
+     | 7 | 7 | seven |   |  0 | 3 | -3
+     | 7 | 7 | seven |   |  0 | 2 |  4
+     | 7 | 7 | seven |   |  0 | 5 | -5
+     | 7 | 7 | seven |   |  0 | 5 | -5
+     | 7 | 7 | seven |   |  0 | 0 |   
+     | 7 | 7 | seven |   |  0 |   |   
      | 7 | 7 | seven |   |  0 |   |  0
-     | 8 | 8 | eight | 1 | -1 |   |  0
-     | 8 | 8 | eight | 2 |  2 |   |  0
-     | 8 | 8 | eight | 3 | -3 |   |  0
-     | 8 | 8 | eight | 2 |  4 |   |  0
-     | 8 | 8 | eight | 5 | -5 |   |  0
-     | 8 | 8 | eight | 5 | -5 |   |  0
-     | 8 | 8 | eight | 0 |    |   |  0
-     | 8 | 8 | eight |   |    |   |  0
+     | 8 | 8 | eight |   |  0 | 1 | -1
+     | 8 | 8 | eight |   |  0 | 2 |  2
+     | 8 | 8 | eight |   |  0 | 3 | -3
+     | 8 | 8 | eight |   |  0 | 2 |  4
+     | 8 | 8 | eight |   |  0 | 5 | -5
+     | 8 | 8 | eight |   |  0 | 5 | -5
+     | 8 | 8 | eight |   |  0 | 0 |   
+     | 8 | 8 | eight |   |  0 |   |   
      | 8 | 8 | eight |   |  0 |   |  0
-     | 0 |   | zero  | 1 | -1 |   |  0
-     | 0 |   | zero  | 2 |  2 |   |  0
-     | 0 |   | zero  | 3 | -3 |   |  0
-     | 0 |   | zero  | 2 |  4 |   |  0
-     | 0 |   | zero  | 5 | -5 |   |  0
-     | 0 |   | zero  | 5 | -5 |   |  0
-     | 0 |   | zero  | 0 |    |   |  0
-     | 0 |   | zero  |   |    |   |  0
+     | 0 |   | zero  |   |  0 | 1 | -1
+     | 0 |   | zero  |   |  0 | 2 |  2
+     | 0 |   | zero  |   |  0 | 3 | -3
+     | 0 |   | zero  |   |  0 | 2 |  4
+     | 0 |   | zero  |   |  0 | 5 | -5
+     | 0 |   | zero  |   |  0 | 5 | -5
+     | 0 |   | zero  |   |  0 | 0 |   
+     | 0 |   | zero  |   |  0 |   |   
      | 0 |   | zero  |   |  0 |   |  0
-     |   |   | null  | 1 | -1 |   |  0
-     |   |   | null  | 2 |  2 |   |  0
-     |   |   | null  | 3 | -3 |   |  0
-     |   |   | null  | 2 |  4 |   |  0
-     |   |   | null  | 5 | -5 |   |  0
-     |   |   | null  | 5 | -5 |   |  0
-     |   |   | null  | 0 |    |   |  0
-     |   |   | null  |   |    |   |  0
+     |   |   | null  |   |  0 | 1 | -1
+     |   |   | null  |   |  0 | 2 |  2
+     |   |   | null  |   |  0 | 3 | -3
+     |   |   | null  |   |  0 | 2 |  4
+     |   |   | null  |   |  0 | 5 | -5
+     |   |   | null  |   |  0 | 5 | -5
+     |   |   | null  |   |  0 | 0 |   
+     |   |   | null  |   |  0 |   |   
      |   |   | null  |   |  0 |   |  0
-     |   | 0 | zero  | 1 | -1 |   |  0
-     |   | 0 | zero  | 2 |  2 |   |  0
-     |   | 0 | zero  | 3 | -3 |   |  0
-     |   | 0 | zero  | 2 |  4 |   |  0
-     |   | 0 | zero  | 5 | -5 |   |  0
-     |   | 0 | zero  | 5 | -5 |   |  0
-     |   | 0 | zero  | 0 |    |   |  0
-     |   | 0 | zero  |   |    |   |  0
+     |   | 0 | zero  |   |  0 | 1 | -1
+     |   | 0 | zero  |   |  0 | 2 |  2
+     |   | 0 | zero  |   |  0 | 3 | -3
+     |   | 0 | zero  |   |  0 | 2 |  4
+     |   | 0 | zero  |   |  0 | 5 | -5
+     |   | 0 | zero  |   |  0 | 5 | -5
+     |   | 0 | zero  |   |  0 | 0 |   
+     |   | 0 | zero  |   |  0 |   |   
      |   | 0 | zero  |   |  0 |   |  0
 (891 rows)