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 |
|:-----------------|----------:|----------:|----------:|----:|----------:|----:|----:|------------:|----------:|-----:|
| ADAIR            |  1389.18  |  6660.43  |  3158.91  |   0 |  356.093  |   0 |   0 |   50.0365   |  3095.38  |  548 |
| ANDREW           |  1705.14  |  4509.9   |  2368.8   |   0 |  160.572  |   0 |   0 |   40.1998   |  1940.33  |  404 |
| ATCHISON         |   642.142 |  2569.91  |   949.319 |   0 |   74.2031 |   0 |   0 |   35.404    |  1510.99  |  146 |
| AUDRAIN          |  2135.34  | 12066.9   |  4963.57  |   0 |  209.111  |   0 |   0 |  232.581    |  6661.64  |  956 |
| BARRY            |  2982.43  | 13005.2   |  7268.08  |   0 |  329.982  |   0 |   0 |   89.6304   |  5317.5   | 1048 |
| BARTON           |  1006.94  |  3899.66  |  1870.65  |   0 |  149.816  |   0 |   0 |   49.6958   |  1829.5   |  339 |
| BATES            |  1503.1   |  6298.95  |  3004.21  |   0 |  145.469  |   0 |   0 |   30.6927   |  3118.57  |  424 |
| BENTON           |  2945.51  | 16100.9   |  8013.11  |   0 |  296.526  |   0 |   0 |  203.255    |  7587.97  |  945 |
| BOLLINGER        |  1329.89  |  6254.47  |  3269.24  |   0 |  122.438  |   0 |   0 |   63.5966   |  2799.19  |  388 |
| BUCHANAN         |  7775.69  | 31270.8   | 13256.4   |   0 | 1781.28   |   0 |   0 |  460.435    | 15772.7   | 2161 |
| CALDWELL         |  1013.71  |  3916.9   |  1711.2   |   0 |   75.9649 |   0 |   0 |    8.0471   |  2121.68  |  311 |
| CALLAWAY         |  4240.78  | 20352.5   |  9531.72  |   0 |  779.21   |   0 |   0 |  169.713    |  9871.87  | 1271 |
| CAMDEN           |  5059.25  | 22646.7   | 10200.4   |   0 |  604.93   |   0 |   0 |  229.697    | 11611.7   | 1673 |
| CAPE GIRARDEAU   |  6481.34  | 24446.7   | 13216.4   |   0 | 1492.55   |   0 |   0 |  305.157    |  9432.59  | 1751 |
| CARROLL          |   883.413 |  4025.69  |  1967.42  |   0 |   93.0206 |   0 |   0 |    6.23882  |  1959.01  |  296 |
| CARTER           |   607.103 |  3697.1   |  1920.96  |   0 |   71.966  |   0 |   0 |    1.13534  |  1703.04  |  196 |
| CASS             |  9570.34  | 34142.5   | 16703.6   |   0 | 2727.42   |   0 |   0 |  472.772    | 14238.7   | 2115 |
| CEDAR            |  1682.4   |  7567.32  |  4042.14  |   0 |   93.6282 |   0 |   0 |   40.0105   |  3391.53  |  612 |
| CHARITON         |   831.83  |  3222.77  |  1538.91  |   0 |   60.2679 |   0 |   0 |   35.5756   |  1588.02  |  283 |
| CHRISTIAN        |  7029.9   | 23407.4   | 14077.4   |   0 | 1979.52   |   0 |   0 |  180.626    |  7169.88  | 1658 |
| CLARK            |   678.731 |  2926.73  |  1410.05  |   0 |   64.2182 |   0 |   0 |   13.2632   |  1439.2   |  203 |
| CLAY             | 19598     | 63067.7   | 29372.2   |   0 | 5161.19   |   0 |   0 |  694.933    | 27839.4   | 3963 |
| CLINTON          |  2047.31  |  8233.16  |  3907.41  |   0 |  319.264  |   0 |   0 |   69.2553   |  3937.24  |  681 |
| COOPER           |  1338.91  |  9036.4   |  3352.93  |   0 |  201.947  |   0 |   0 |   53.1945   |  5428.33  |  535 |
| CRAWFORD         |  2786.59  | 10004.4   |  5143     |   0 |  323.958  |   0 |   0 |   35.7316   |  4501.73  |  681 |
| DADE             |   842.943 |  3980.83  |  2016.36  |   0 |  101.068  |   0 |   0 |   70.4375   |  1792.96  |  291 |
| DALLAS           |  1753.87  |  7775.87  |  4680.86  |   0 |  174.927  |   0 |   0 |   30.1173   |  2889.97  |  462 |
| DAVIESS          |   821.573 |  3499.8   |  1672.77  |   0 |  108.975  |   0 |   0 |    8.35798  |  1709.7   |  271 |
| DE KALB          |  1465.72  |  4301.52  |  1684.24  |   0 |  188.723  |   0 |   0 |   42.7565   |  2385.8   |  281 |
| DENT             |  1611.39  |  8244.58  |  4953.8   |   0 |  237.374  |   0 |   0 |   24.5804   |  3028.82  |  433 |
| DOUGLAS          |  1258.19  |  3852.61  |  3110.61  |   0 |  122.302  |   0 |   0 |   26.5395   |   593.159 |  107 |
| DUNKLIN          |  2262.66  | 13699.3   |  7160.74  |   0 |  239.091  |   0 |   0 |   32.805    |  6266.67  |  835 |
| FRANKLIN         |  8040.29  | 24988.7   | 13042.6   |   0 | 1595.82   |   0 |   0 |  560.554    |  9789.78  | 1513 |
| GASCONADE        |  1720.71  |  5479.64  |  3005.75  |   0 |   89.3174 |   0 |   0 |   73.0757   |  2311.49  |  434 |
| GENTRY           |   602.343 |  2345     |  1117     |   0 |   68.5766 |   0 |   0 |   24.1243   |  1135.3   |  178 |
| GRUNDY           |   789.247 |  3802.54  |  1688.86  |   0 |  161.016  |   0 |   0 |   40.4362   |  1912.22  |  312 |
| HARRISON         |   870.951 |  3234.36  |  1418.97  |   0 |   40.1804 |   0 |   0 |  128.152    |  1647.05  |  265 |
| HENRY            |  2562.05  | 11262.5   |  6118.18  |   0 |  389.249  |   0 |   0 |   88.4406   |  4666.63  |  773 |
| HICKORY          |  1312.72  |  5268.85  |  3096.68  |   0 |   21.3421 |   0 |   0 |   38.238    |  2112.58  |  352 |
| HOLT             |   558.833 |  1819.38  |   723.625 |   0 |   55.1581 |   0 |   0 |   53.7878   |   986.811 |  144 |
| HOWARD           |   830.23  |  4678.63  |  2145.99  |   0 |   86.0616 |   0 |   0 |   91.7986   |  2354.79  |  305 |
| HOWELL           |  4097.49  | 19158.6   | 10960.8   |   0 |  707.045  |   0 |   0 |   37.1227   |  7453.66  | 1386 |
| IRON             |  1216.81  |  6034.96  |  3240.72  |   0 |   82.8033 |   0 |   0 |  145.664    |  2565.77  |  334 |
| JASPER           |  9107.58  | 36320.9   | 18957.1   |   0 | 2050.57   |   0 |   0 |  480.384    | 14832.8   | 2944 |
| JEFFERSON        | 19654.9   | 59955.9   | 28469.2   |   0 | 4614.76   |   0 |   0 | 1084.64     | 25787.3   | 3385 |
| JOHNSON          |  5817.49  | 25835.2   | 14343.9   |   0 | 4090.3    |   0 |   0 |  283.293    |  7117.64  | 1217 |
| LACLEDE          |  3381.17  | 18514.3   | 11439.5   |   0 | 1076.85   |   0 |   0 |  234.203    |  5763.78  |  982 |
| LAFAYETTE        |  3208.1   | 11862.3   |  6102.18  |   0 |  628.201  |   0 |   0 |  204.277    |  4927.69  |  873 |
| LAWRENCE         |  3222.12  | 16776.2   |  8658.4   |   0 |  450.682  |   0 |   0 |  165.355    |  7501.78  | 1556 |
| LEWIS            |   906.462 |  2844.96  |  1222.43  |   0 |  160.711  |   0 |   0 |   68.4433   |  1393.37  |  247 |
| LINCOLN          |  4149.45  | 12069.6   |  5973.82  |   0 |  786.176  |   0 |   0 |  279.513    |  5030.08  |  686 |
| LINN             |  1117.82  |  6000.57  |  2425.52  |   0 |  125.151  |   0 |   0 |   57.1683   |  3392.74  |  508 |
| LIVINGSTON       |  1349.17  |  4919.68  |  2632.96  |   0 |  115.333  |   0 |   0 |   36.8043   |  2134.59  |  461 |
| MC DONALD        |  1884.54  |  6487.42  |  3166.72  |   0 |  184.255  |   0 |   0 |   52.5783   |  3083.86  |  501 |
| MACON            |  1386.5   |  7967.47  |  3190.37  |   0 |  167.637  |   0 |   0 |   22.3584   |  4587.11  |  580 |
| MADISON          |  1327.02  |  7186.82  |  3631.86  |   0 |  143.147  |   0 |   0 |   73.4353   |  3338.38  |  450 |
| MARIES           |   949.764 |  5208.06  |  3319.58  |   0 |  195.929  |   0 |   0 |   27.9645   |  1664.59  |  266 |
| MARION           |  2527.76  |  8749.43  |  4347.24  |   0 |  337.37   |   0 |   0 |  249.998    |  3814.82  |  684 |
| MERCER           |   406.105 |  1682.44  |   783.729 |   0 |   43.7067 |   0 |   0 |   15.6882   |   839.319 |  133 |
| MILLER           |  2376.22  | 10109.3   |  5091.7   |   0 |  356.451  |   0 |   0 |   87.7695   |  4573.41  |  815 |
| MISSISSIPPI      |   870.007 |  4316.49  |  1927.62  |   0 |  100.58   |   0 |   0 |   82.9034   |  2205.38  |  369 |
| MONITEAU         |  1434.13  |  4057.68  |  2112.92  |   0 |  140.223  |   0 |   0 |   18.8731   |  1785.66  |  336 |
| MONROE           |   920.483 |  4355.09  |  2022.83  |   0 |   78.8168 |   0 |   0 |   42.863    |  2210.58  |  290 |
| MONTGOMERY       |  1114.58  |  4761.29  |  1685.37  |   0 |  137.566  |   0 |   0 |   21.885    |  2916.47  |  375 |
| MORGAN           |  2263.17  | 12866.8   |  6242.74  |   0 |  218.812  |   0 |   0 |   88.8868   |  6316.38  |  913 |
| NEW MADRID       |  1652.18  |  6900.47  |  2788.46  |   0 |  148.709  |   0 |   0 |   38.2704   |  3925.03  |  529 |
| NEWTON           |  5470.92  | 20603.4   | 10710.5   |   0 |  989.347  |   0 |   0 |  198.651    |  8704.93  | 1641 |
| NODAWAY          |  1511.77  |  5548.97  |  2692.13  |   0 |  484.554  |   0 |   0 |   31.3357   |  2340.96  |  345 |
| OREGON           |  1299.84  |  5870.45  |  3568.52  |   0 |  104.964  |   0 |   0 |   29.0435   |  2167.92  |  348 |
| OSAGE            |  1067.77  |  4024.64  |  1982.1   |   0 |  199.359  |   0 |   0 |   66.196    |  1776.99  |  310 |
| OZARK            |  1180.02  |  4488.81  |  2791.87  |   0 |  135.235  |   0 |   0 |   18.9492   |  1542.76  |  271 |
| PEMISCOT         |  1258.92  |  6791.85  |  3572.47  |   0 |  159.515  |   0 |   0 |   89.0239   |  2970.85  |  372 |
| PERRY            |  1680.96  |  5009.58  |  2995.51  |   0 |  299.398  |   0 |   0 |  228.15     |  1486.53  |  399 |
| PETTIS           |  3485.35  | 19128.8   |  9599.85  |   0 | 1173.72   |   0 |   0 |  217.513    |  8137.76  | 1252 |
| PHELPS           |  4644.92  | 24418.2   | 14530.8   |   0 | 1550.19   |   0 |   0 |  264.516    |  8072.74  | 1335 |
| PIKE             |  1784.92  |  6033.23  |  2680.69  |   0 |  163.829  |   0 |   0 |  136.451    |  3052.26  |  401 |
| PLATTE           |  7037.5   | 29179.3   | 13613.1   |   0 | 2534      |   0 |   0 |  310.829    | 12721.4   | 1627 |
| POLK             |  2903.72  | 11531.4   |  5674.06  |   0 |  477.48   |   0 |   0 |  168.514    |  5211.35  |  858 |
| PULASKI          |  5906.94  | 58845.1   | 38681.8   |   0 | 8484.32   |   0 |   0 |  198.785    | 11480.2   | 2084 |
| PUTNAM           |   502.246 |  1990.29  |   849.086 |   0 |   39.0079 |   0 |   0 |    0.508521 |  1101.68  |  163 |
| RANDOLPH         |  2370.09  | 12166.8   |  4585.04  |   0 |  362.96   |   0 |   0 |  139.865    |  7078.97  |  669 |
| REYNOLDS         |   744.974 |  4986.47  |  2534.32  |   0 |  104.522  |   0 |   0 |   21.9675   |  2325.66  |  276 |
| RIPLEY           |  1447.5   | 10751.5   |  5244.89  |   0 |  140.652  |   0 |   0 |   13.2589   |  5352.69  |  610 |
| ST. CHARLES      | 30438.4   | 68795     | 38311.1   |   0 | 7565.27   |   0 |   0 | 1597.01     | 21321.7   | 3403 |
| ST. CLAIR        |  1291.36  |  6352.39  |  3166.34  |   0 |  125.04   |   0 |   0 |   35.8805   |  3025.13  |  358 |
| SAINTE GENEVIEVE |  1833.46  |  7384.27  |  3925.91  |   0 |  195.803  |   0 |   0 |   79.179    |  3183.38  |  478 |
| ST. FRANCOIS     |  6113.22  | 27031.7   | 12527.1   |   0 |  923.313  |   0 |   0 |  573.935    | 13007.3   | 1779 |
| SALINE           |  1720.09  |  8080.83  |  3652.58  |   0 |  291.726  |   0 |   0 |  102.032    |  4034.5   |  540 |
| SCHUYLER         |   309.628 |  1415.41  |   538.736 |   0 |   17.7896 |   0 |   0 |   18.4632   |   840.42  |  130 |
| SCOTT            |  3525.27  | 16246.9   |  9076.01  |   0 |  423.676  |   0 |   0 |   94.0189   |  6653.17  | 1075 |
| SHANNON          |   899.567 |  4783.65  |  2323.94  |   0 |  111.192  |   0 |   0 |    0.783531 |  2347.74  |  289 |
| SHELBY           |   511.846 |  2539.39  |  1080.05  |   0 |   85.9537 |   0 |   0 |   90.6687   |  1282.71  |  214 |
| STODDARD         |  2798.61  | 17013.4   |  8736.85  |   0 |  305.687  |   0 |   0 |   89.1301   |  7881.71  | 1083 |
| STONE            |  4058.72  | 16186.7   |  8291.42  |   0 |  349.386  |   0 |   0 |  228.02     |  7317.85  | 1432 |
| SULLIVAN         |   540.398 |  2201.93  |  1255.81  |   0 |   54.4897 |   0 |   0 |    2.87902  |   888.749 |  180 |
| TANEY            |  4472.61  | 22869.6   | 10979.5   |   0 |  626.29   |   0 |   0 |  307.243    | 10956.6   | 2092 |
| TEXAS            |  2927.05  | 17781.4   | 10866.9   |   0 |  935.772  |   0 |   0 |  118.481    |  5860.23  |  975 |
| VERNON           |  1447.64  |  8200.45  |  3697.97  |   0 |  263.984  |   0 |   0 |  123.296    |  4115.2   |  729 |
| WARREN           |  2699.33  |  8281.52  |  4545.39  |   0 |  535.481  |   0 |   0 |  132.264    |  3068.38  |  515 |
| WASHINGTON       |  1709.33  |  8178.19  |  4408.07  |   0 |  183.961  |   0 |   0 |   54.8721   |  3531.29  |  501 |
| WAYNE            |  1514.9   | 10011.1   |  4439.83  |   0 |  118.264  |   0 |   0 |   32.7566   |  5420.21  |  499 |
| WEBSTER          |  3696.75  | 11997.2   |  6968.29  |   0 |  522.415  |   0 |   0 |  114.563    |  4391.9   |  819 |
| WORTH            |   210.255 |   898.525 |   402.432 |   0 |   21.4193 |   0 |   0 |    1.36902  |   473.305 |   72 |
| WRIGHT           |  1624.36  |  9850.18  |  5824.58  |   0 |  303.997  |   0 |   0 |   99.6434   |  3621.96  |  531 |



labeled_formulas: G=B*E
H=B*E
H=B*G
G=C*E
H=C*E
H=C*G
G=D*E
H=D*E
H=D*G
G=E*F
H=E*F
G=E*H
G=E*I
G=E*J
G=E*K
H=E*I
H=E*J
H=E*K
H=F*G
H=G*I
H=G*J
H=G*K
G=(B+C)*E
H=(B+C)*E
H=(B+C)*G
G=(B+D)*E
H=(B+D)*E
H=(B+D)*G
G=(B*E)*F
H=(B*E)*F
G=(B*E)*H
G=(B*E)*I
G=(B*E)*J
G=(B*E)*K
H=(B*E)*I
H=(B*E)*J
H=(B*E)*K
H=(B+F)*G
H=(B*G)*I
H=(B*G)*J
H=(B*G)*K
G=(C+D)*E
H=(C+D)*E
H=(C+D)*G
G=(C*E)*F
H=(C*E)*F
G=(C*E)*H
G=(C*E)*I
G=(C*E)*J
G=(C*E)*K
H=(C*E)*I
H=(C*E)*J
H=(C*E)*K
H=(C+F)*G
H=(C*G)*I
H=(C*G)*J
H=(C*G)*K
G=(D*E)*F
H=(D*E)*F
G=(D*E)*H
G=(D*E)*I
G=(D*E)*J
G=(D*E)*K
H=(D*E)*I
H=(D*E)*J
H=(D*E)*K
H=(D+F)*G
H=(D*G)*I
H=(D*G)*J
H=(D*G)*K
G=(E*F)*H
G=(E*F)*I
output: {"Arithmetic-Relationship": []}
source: $MMTU_HOME/resources/Relationship-finding/AR/table_119124___xlsx_41733d0294b45d708bccba64327d3a7c1798edcb.xlsx___line_188