Task: Arithmetic-Relationship (source dataset: Auto-Relate)

You are given a table with numerical values in multiple columns. Your task is to analyze the data and identify semantic relationships, expressed in arithmetic relationships, that exist between columns.

These relationships may involve any combination of the following operations: addition (+), subtraction (−), multiplication (*), and division (/). The expressions can involve two or more columns and may be nested (e.g., B = (A + C) / D).

Please only identify arithmetic relationships that hold semantically based on the data, and do not return relationships that are likely spurious, e.g., (A = B * 0), where A is an all 0 column. Please only return minimal atomic relationships, e.g., if A = B + C and D = E + F, then there is no need to return A = B + C + E + F - D.

No explanation, return all the formula(s) in JSON format: {"Arithmetic-Relationship": [<list-of-formulas>]}. If no relationship is found, return {"Arithmetic-Relationship": []}.

Input Table:
|     A | B                        |        C |         D |           E |        F |     G | H   | I   |   J |   K |
|------:|:-------------------------|---------:|----------:|------------:|---------:|------:|:----|:----|----:|----:|
|  6001 | Aberdeen 06-1            |  4469.94 |  419.722  |  10.6498    |  8.1     | 1.512 | Yes | No  |   0 |   0 |
| 61001 | Alcester-Hudson 61-1     |   338.39 |  194.248  |   1.74205   |  8.2     | 1.761 | Yes | No  |   0 |   0 |
| 11001 | Andes Central 11-1       |   316    |  204.286  |   1.54685   | 13.195   | 2.161 | Yes | No  |   0 |   0 |
| 38001 | Arlington 38-1           |   256    |  231.548  |   1.1056    | 10.9     | 1.88  | Yes | No  |   0 |   0 |
| 21001 | Armour 21-1              |   179    |  129.511  |   1.38212   |  7.6     | 2.982 | Yes | No  |   0 |   0 |
|  4001 | Avon 04-1                |   232    |  179.456  |   1.29279   |  9.7     | 1.512 | Yes | No  |   0 |   0 |
| 49001 | Baltic 49-1              |   479    |   54.2802 |   8.82458   |  5.1     | 1.512 | Yes | No  |   0 |   0 |
|  9001 | Belle Fourche 09-1       |  1361.33 |  956.931  |   1.4226    | 11.4     | 1.512 | Yes | No  |   0 |   0 |
| 61002 | Beresford 61-2           |   693.33 |  205.093  |   3.38056   |  8.2     | 1.95  | Yes | No  |   0 |   0 |
| 25001 | Big Stone City 25-1      |    91    |   20.6562 |   4.40546   |  0       | 2.432 | No  | No  |   0 |   0 |
|  4002 | Bon Homme 04-2           |   531    |  316.113  |   1.67978   |  9.7     | 1.512 | Yes | No  |   0 |   0 |
| 22001 | Bowdle 22-1              |   109    |  274.912  |   0.396491  | 15.2     | 2.474 | Yes | No  |   0 |   0 |
| 49002 | Brandon Valley 49-2      |  4249.75 |  126.204  |  33.6737    |  8.5     | 1.512 | Yes | No  |   0 |   0 |
| 30003 | Bridgewater-Emery 30-3   |   340    |  229.816  |   1.47944   |  9.1     | 1.948 | Yes | No  |   0 |   0 |
| 45004 | Britton-Hecla 45-4       |   418.75 |  661.075  |   0.633438  | 13.8     | 1.512 | Yes | No  |   0 |   0 |
|  5001 | Brookings 05-1           |  3402.03 |  193.82   |  17.5525    |  7.1     | 1.66  | Yes | No  |   0 |   0 |
| 26002 | Burke 26-2               |   243    |  350.465  |   0.693365  |  7.9     | 2.341 | Yes | No  |   0 |   0 |
| 43001 | Canistota 43-1           |   193    |   98.489  |   1.95961   |  8.8     | 2.425 | Yes | No  |   0 |   0 |
| 41001 | Canton 41-1              |   872.88 |  194.101  |   4.49705   | 10.1     | 1.512 | Yes | No  |   0 |   0 |
| 28001 | Castlewood 28-1          |   294    |  129.871  |   2.26379   | 11.7     | 1.512 | Yes | No  |   0 |   0 |
| 60001 | Centerville 60-1         |   273.39 |  138.778  |   1.96998   |  7.3     | 1.512 | Yes | No  |   0 |   0 |
|  7001 | Chamberlain 07-1         |   885.51 |  928.683  |   0.953512  | 18.7     | 1.512 | Yes | No  |   0 |   0 |
| 39001 | Chester 39-1             |   531    |  140.936  |   3.76767   |  8.5     | 1.512 | Yes | No  |   0 |   0 |
| 12002 | Clark 12-2               |   376    |  623.976  |   0.602587  | 13.7     | 1.785 | Yes | No  |   0 |   0 |
| 50005 | Colman-Egan 50-5         |   252.6  |  161.137  |   1.56761   |  8.5     | 1.512 | Yes | No  |   0 |   0 |
| 59003 | Colome Consolidated 59-3 |   229    |  806.477  |   0.283951  | 10.1     | 1.512 | Yes | No  |   0 |   0 |
| 21003 | Corsica-Stickney 21-3    |   253    |  382.463  |   0.661502  |  7.6     | 1.512 | Yes | No  |   0 |   0 |
| 16001 | Custer 16-1              |   958.86 | 1207.73   |   0.793933  | 11.2     | 1.512 | Yes | No  |   0 |   0 |
| 61008 | Dakota Valley 61-8       |  1355.41 |   29.4881 |  45.9647    | 12.7     | 1.742 | Yes | No  |   0 |   0 |
| 38002 | De Smet 38-2             |   285    |  312.602  |   0.911701  |  9.1     | 1.78  | Yes | No  |   0 |   0 |
| 49003 | Dell Rapids 49-3         |   951.27 |  168.108  |   5.65869   |  5.1     | 1.512 | Yes | No  |   0 |   0 |
|  5006 | Deubrook 05-6            |   379    |  250.386  |   1.51366   | 11.4     | 1.704 | Yes | No  |   0 |   0 |
| 19004 | Deuel 19-4               |   513.25 |  474.266  |   1.0822    | 16.1     | 1.512 | Yes | No  |   0 |   0 |
| 51001 | Douglas 51-1             |  2907    |  151      |  19.2517    |  8.6     | 1.512 | Yes | No  |   0 |   0 |
| 16002 | Elk Mountain 16-2        |    13    |  310.196  |   0.0419089 |  0       | 1.512 | Yes | No  |   0 |   0 |
| 61007 | Elk Point-Jefferson 61-7 |   687    |  215.97   |   3.181     | 12.7     | 1.512 | Yes | No  |   0 |   0 |
|  5003 | Elkton 05-3              |   322    |  150.122  |   2.14492   | 13.9     | 1.512 | Yes | No  |   0 |   0 |
| 28002 | Estelline 28-2           |   261    |  169.401  |   1.54072   | 12.1     | 2.235 | Yes | No  |   0 |   0 |
| 17001 | Ethan 17-1               |   269.8  |  105.479  |   2.55786   | 10.5     | 1.512 | Yes | No  |   0 |   0 |
| 50003 | Flandreau 50-3           |   690.28 |  224.653  |   3.07265   | 11.3     | 1.512 | Yes | No  |   0 |   0 |
| 14001 | Florence 14-1            |   257    |  140.233  |   1.83267   | 13.7     | 1.512 | Yes | No  |   0 |   0 |
|  6002 | Frederick Area 06-2      |   163    |  351.415  |   0.463839  | 22.2     | 1.855 | Yes | No  |   0 |   0 |
| 33001 | Freeman 33-1             |   320.03 |  238.825  |   1.34002   | 10.1     | 2.741 | Yes | No  |   0 |   0 |
| 49004 | Garretson 49-4           |   480.43 |   88.3728 |   5.4364    |  9.1     | 2.343 | Yes | No  |   0 |   0 |
| 63001 | Gayville-Volin 63-1      |   293    |   72.3129 |   4.05184   | 11.927   | 1.512 | Yes | No  |   0 |   0 |
| 53001 | Gettysburg 53-1          |   239.04 |  222.428  |   1.07468   | 18.3     | 1.512 | Yes | No  |   0 |   0 |
| 26004 | Gregory 26-4             |   373.6  |  515.453  |   0.724799  |  7.9     | 1.81  | Yes | No  |   0 |   0 |
|  6006 | Groton Area 06-6         |   578.87 |  872.336  |   0.663586  | 16.7     | 1.512 | Yes | No  |   0 |   0 |
| 28003 | Hamlin 28-3              |   810    |  364.244  |   2.22379   | 11.7     | 1.512 | Yes | No  |   0 |   0 |
| 30001 | Hanson 30-1              |   402    |  257.485  |   1.56126   |  9.1     | 1.512 | Yes | No  |   0 |   0 |
| 41002 | Harrisburg 41-2          |  4807.77 |   70.9566 |  67.7565    |  6.7     | 1.769 | Yes | No  |   0 |   0 |
| 14002 | Henry 14-2               |   176    |  100.19   |   1.75667   | 13.7     | 3.6   | Yes | No  |   0 |   0 |
| 10001 | Herreid 10-1             |   119    |  274.059  |   0.434213  | 22       | 2.267 | Yes | No  |   0 |   0 |
| 51002 | Hill City 51-2           |   453.4  |  579.579  |   0.782292  | 11.2     | 1.512 | Yes | No  |   0 |   0 |
| 56006 | Hitchcock-Tulare 56-6    |   230.38 |  483.423  |   0.47656   |  9.6     | 1.839 | Yes | No  |   0 |   0 |
| 23002 | Hot Springs 23-2         |   761.24 |  591.1    |   1.28784   | 19.9     | 1.512 | Yes | No  |   0 |   0 |
| 48003 | Howard 48-3              |   363.1  |  530.906  |   0.683925  | 17.1     | 1.512 | Yes | No  |   0 |   0 |
|  2002 | Huron 02-2               |  2660.62 |  433.139  |   6.14264   | 12.9     | 1.512 | Yes | No  |   0 |   0 |
| 22006 | Ipswich Public 22-6      |   422.49 |  534.705  |   0.790137  | 15       | 1.512 | Yes | No  |   0 |   0 |
| 13003 | Irene-Wakonda 13-3       |   283.72 |  283.608  |   1.0004    |  6.94615 | 1.811 | Yes | No  |   0 |   0 |
|  2003 | Iroquois 02-3            |   223.2  |  376.411  |   0.592969  | 14.8     | 2.433 | Yes | No  |   0 |   0 |
|  7002 | Kimball 07-2             |   305.25 |  474.21   |   0.643702  | 12.5     | 1.512 | Yes | No  |   0 |   0 |
| 38003 | Lake Preston 38-3        |   164    |  198.07   |   0.827989  |  9.1     | 2.89  | Yes | No  |   0 |   0 |
| 45005 | Langford Area 45-5       |   211    |  424.799  |   0.496705  | 13.8     | 1.512 | Yes | No  |   0 |   0 |
| 40001 | Lead-Deadwood 40-1       |   757.99 |  431.899  |   1.75502   |  6.1     | 1.512 | Yes | No  |   0 |   0 |
| 41004 | Lennox 41-4              |  1123.75 |  191.831  |   5.85803   |  7.2     | 1.512 | Yes | No  |   0 |   0 |
| 39002 | Madison Central 39-2     |  1205.8  |  250.911  |   4.80569   |  8.5     | 1.558 | Yes | No  |   0 |   0 |
| 60003 | Marion 60-3              |   167    |  110.279  |   1.51434   |  6.2     | 1.512 | Yes | No  |   0 |   0 |
| 43007 | McCook Central 43-7      |   377.91 |  221.978  |   1.70246   | 10.1     | 2.015 | Yes | No  |   0 |   0 |
| 15002 | McLaughlin 15-2          |   444.87 |  794.558  |   0.559896  | 17       | 1.512 | Yes | No  |   0 |   0 |
| 46001 | Meade 46-1               |  2878.35 | 3124.99   |   0.921075  | 13.3     | 1.512 | Yes | No  |   0 |   0 |
| 33002 | Menno 33-2               |   277    |  179.491  |   1.54326   |  9.2     | 1.512 | Yes | No  |   0 |   0 |
| 25004 | Milbank 25-4             |   987.2  |  489.16   |   2.01815   | 17.29    | 1.512 | Yes | No  |   0 |   0 |
| 17002 | Mitchell 17-2            |  2795.95 |  265.93   |  10.5139    | 12       | 1.512 | Yes | No  |   0 |   0 |
| 62006 | Mobridge-Pollock 62-6    |   618.42 |  266.429  |   2.32114   |  9.3     | 1.791 | Yes | No  |   0 |   0 |
| 43002 | Montrose 43-2            |   239    |  124.286  |   1.92299   |  8.8     | 1.512 | Yes | No  |   0 |   0 |
| 17003 | Mount Vernon 17-3        |   213    |  167.692  |   1.27018   | 11.1     | 2.127 | Yes | No  |   0 |   0 |
| 51003 | New Underwood 51-3       |   237    |  356      |   0.665731  | 11.7     | 1.512 | Yes | No  |   0 |   0 |
| 56007 | Northwestern Area 56-7   |   266    |  669.13   |   0.397531  | 11.6     | 1.512 | Yes | No  |   0 |   0 |
| 39005 | Oldham-Ramona 39-5       |   170    |  191.867  |   0.886029  |  9.3     | 2.06  | Yes | No  |   0 |   0 |
| 60004 | Parker 60-4              |   437    |  137.915  |   3.16863   |  6.2     | 1.512 | Yes | No  |   0 |   0 |
| 33003 | Parkston 33-3            |   534    |  307.298  |   1.73773   | 10.5     | 1.512 | Yes | No  |   0 |   0 |
| 32002 | Pierre 32-2              |  2669    |  355.186  |   7.51437   |  1.7     | 1.512 | Yes | No  |   0 |   0 |
|  1001 | Plankinton 01-1          |   341    |  277.183  |   1.23024   | 11.01    | 1.921 | Yes | No  |   0 |   0 |
| 11005 | Platte-Geddes 11-5       |   503.37 |  631.324  |   0.797325  | 21.7697  | 1.512 | Yes | No  |   0 |   0 |
| 51004 | Rapid City Area 51-4     | 13679.7  |  420.313  |  32.5464    |  1.1     | 1.512 | Yes | No  |   0 |   0 |
| 56004 | Redfield 56-4            |   591.65 |  411.779  |   1.43681   |  9.6     | 1.512 | Yes | No  |   0 |   0 |
| 54004 | Rosholt 54-4             |   249    |  173.358  |   1.43633   | 20.8     | 2.024 | Yes | No  |   0 |   0 |
| 39004 | Rutland 39-4             |   187    |  125.103  |   1.49477   |  8.5     | 3.234 | Yes | No  |   0 |   0 |
| 55005 | Sanborn Central 55-5     |   189    |  395.4    |   0.477997  |  8.8     | 2.756 | Yes | No  |   0 |   0 |
|  4003 | Scotland 04-3            |   253    |  257.799  |   0.981385  |  9.2     | 1.696 | Yes | No  |   0 |   0 |
| 65001 | Shannon County 65-1      |  1339.16 | 2099.17   |   0.637946  |  0       | 1.512 | Yes | No  |   0 |   0 |
| 49005 | Sioux Falls 49-5         | 24024.8  |   74.4137 | 322.854     |  8.5     | 1.747 | Yes | No  |   0 |   0 |
|  5005 | Sioux Valley 05-5        |   682.67 |  185.057  |   3.68897   |  7.1     | 1.887 | Yes | No  |   0 |   0 |
| 54002 | Sisseton 54-2            |   897    |  851.281  |   1.05371   | 20.4     | 1.512 | Yes | No  |   0 |   0 |
| 15003 | Smee 15-3                |   179    |  200.381  |   0.893296  |  9.3     | 1.512 | Yes | No  |   0 |   0 |
| 26005 | South Central 26-5       |    86    |  316.052  |   0.272107  | 18.5933  | 2.45  | Yes | No  |   0 |   0 |
| 40002 | Spearfish 40-2           |  2390.07 |  283.882  |   8.41923   | 11.2     | 1.512 | Yes | No  |   0 |   0 |
| 57001 | Stanley County 57-1      |   435.86 | 1516.58   |   0.287397  |  1.7     | 1.512 | Yes | No  |   0 |   0 |
| 54006 | Summit 54-6              |   158    |  156.532  |   1.00938   | 11.3     | 2.329 | Yes | No  |   0 |   0 |
| 41005 | Tea Area 41-5            |  1906.5  |   22.589  |  84.3996    |  6.7     | 1.512 | Yes | No  |   0 |   0 |
| 66001 | Todd County 66-1         |  2106.8  | 1390.39   |   1.51526   | 19       | 1.512 | Yes | No  |   0 |   0 |
| 33005 | Tripp-Delmont 33-5       |   130    |  250.878  |   0.518179  | 11.5     | 2.327 | Yes | No  |   0 |   0 |
| 49006 | Tri-Valley 49-6          |   968    |  147.609  |   6.55788   |  7.8     | 2.008 | Yes | No  |   0 |   0 |
| 13001 | Vermillion 13-1          |  1259.26 |  179.068  |   7.03231   | 13.4     | 1.976 | Yes | No  |   0 |   0 |
| 60006 | Viborg-Hurley 60-6       |   346    |  206.538  |   1.67524   |  6.9     | 1.512 | Yes | No  |   0 |   0 |
| 11004 | Wagner 11-4              |   839    |  329.683  |   2.54487   | 13.2     | 2.029 | Yes | No  |   0 |   0 |
|  6005 | Warner 06-5              |   310    |  188.986  |   1.64033   |  8.1     | 1.512 | Yes | No  |   0 |   0 |
| 14004 | Watertown 14-4           |  3927.97 |  326.16   |  12.0431    |  9.5     | 1.512 | Yes | No  |   0 |   0 |
| 18003 | Waubay 18-3              |   170    |  198.995  |   0.854291  | 10.2     | 2.274 | Yes | No  |   0 |   0 |
| 14005 | Waverly 14-5             |   235    |  250.55   |   0.937936  |  9.2     | 2.153 | Yes | No  |   0 |   0 |
| 18005 | Webster Area 18-5        |   542    |  491.952  |   1.10173   | 10.2     | 1.512 | Yes | No  |   0 |   0 |
| 36002 | Wessington Springs 36-2  |   312.18 |  745.669  |   0.418657  | 14.7     | 1.77  | Yes | No  |   0 |   0 |
| 49007 | West Central 49-7        |  1410.25 |  165.116  |   8.54095   |  7.8     | 1.763 | Yes | No  |   0 |   0 |
|  1003 | White Lake 01-3          |   119    |  245.779  |   0.484175  | 11.2     | 2.021 | Yes | No  |   0 |   0 |
| 12003 | Willow Lake 12-3         |   249    |  301.871  |   0.824855  | 17.2     | 1.512 | Yes | No  |   0 |   0 |
| 54007 | Wilmot 54-7              |   223    |  224.894  |   0.99158   | 11.3     | 1.512 | Yes | No  |   0 |   0 |
| 59002 | Winner 59-2              |   710    | 1184.79   |   0.59926   | 10.1     | 1.512 | Yes | No  |   0 |   0 |
|  2006 | Wolsey-Wessington 02-6   |   346    |  401.792  |   0.861142  | 12.9     | 1.953 | Yes | No  |   0 |   0 |
| 55004 | Woonsocket 55-4          |   248    |  219.968  |   1.12744   |  8.8     | 1.671 | Yes | No  |   0 |   0 |
| 63003 | Yankton 63-3             |  2775.69 |  216.072  |  12.8461    | 11.8     | 1.512 | Yes | No  |   0 |   0 |



label: E=C/D
labeled_formulas: E=C/D
K=A*J
K=C*J
K=D*J
K=E*J
K=F*J
K=G*J
K=(A+C)*J
K=(A+D)*J
K=(A+E)*J
K=(A+F)*J
K=(A+G)*J
E=(C/D)+J
E=(C/D)+K
K=(C+D)*J
K=(C+E)*J
K=(C+F)*J
K=(C+G)*J
K=(D+E)*J
K=(D+F)*J
K=(D+G)*J
K=(E+F)*J
K=(E+G)*J
K=(F+G)*J
output: {"Arithmetic-Relationship": ["E=C/D"]}
source: $MMTU_HOME/resources/Relationship-finding/AR/table_117415___xlsx_92e0bab050eefd6b182fd2f400b83f9965132b39.xlsx___line_294