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 |
|:-------------|---------:|----------:|---------:|----:|-----------:|----:|----:|-----------:|-----------:|------:|
| ALLEGANY     |  4452.29 |  22574.3  |  9953.23 |   0 |   824.753  |   0 |   0 |   148.448  |  11647.9   |  1514 |
| BROOME       | 14867.2  |  46110.3  | 25181.9  |   0 |  3245.03   |   0 |   0 |  1149.94   |  16533.4   |  3434 |
| CATTARAUGUS  |  7753.55 |  34804.5  | 18176.2  |   0 |  1235.4    |   0 |   0 |   515.098  |  14877.7   |  2259 |
| CAYUGA       |  6596.25 |  26760.7  | 12374.5  |   0 |  1210      |   0 |   0 |   317.966  |  12858.2   |  1890 |
| CHAUTAUQUA   | 11250.3  |  51444    | 24347.7  |   0 |  2512.14   |   0 |   0 |   910.264  |  23673.8   |  3960 |
| CHEMUNG      |  7743.22 |  41221.3  | 18233.6  |   0 |  1658.32   |   0 |   0 |   822.27   |  20507.1   |  2614 |
| CHENANGO     |  4620.36 |  17289.9  |  9218.53 |   0 |   707.126  |   0 |   0 |   141.635  |   7222.63  |  1028 |
| CLINTON      |  6886.68 |  28353.1  | 17771.9  |   0 |  1424.93   |   0 |   0 |   552.084  |   8604.14  |  1993 |
| COLUMBIA     |  5481.72 |  26185    |  7120.33 |   0 |  5408.25   |   0 |   0 |   417.739  |  13238.7   |  1187 |
| CORTLAND     |  3368.34 |  14520.8  |  5464.03 |   0 |   657.328  |   0 |   0 |   196.709  |   8202.76  |  1082 |
| DELAWARE     |  4128.61 |  13282.7  |  6598.95 |   0 |   784.8    |   0 |   0 |   307.984  |   5590.94  |   855 |
| ESSEX        |  4108.29 |  12260.7  |  6781.3  |   0 |   379.024  |   0 |   0 |   326.75   |   4773.66  |   885 |
| FRANKLIN     |  4042.56 |  14744.3  |  8361.41 |   0 |   671.019  |   0 |   0 |   218.9    |   5492.97  |  1012 |
| FULTON       |  4837.97 |  16949.5  |  8369.82 |   0 |   919.142  |   0 |   0 |   255.578  |   7404.92  |  1021 |
| GREENE       |  4219.89 |  23100.9  |  7644.13 |   0 |  1515.58   |   0 |   0 |   300.55   |  13640.7   |  1166 |
| HAMILTON     |   637.74 |   1180.48 |   637.4  |   0 |    27.7094 |   0 |   0 |    39.2427 |    476.126 |    87 |
| HERKIMER     |  5810.59 |  20580.3  | 11530.7  |   0 |   813.982  |   0 |   0 |   239.706  |   7995.94  |  1235 |
| JEFFERSON    | 10748.7  |  57764.8  | 31868.4  |   0 |  8843.84   |   0 |   0 |   744.09   |  16308.5   |  3214 |
| LEWIS        |  2386.31 |   8738.22 |  4873.79 |   0 |   433.117  |   0 |   0 |   286.754  |   3144.56  |   649 |
| LIVINGSTON   |  5138.08 |  17962.8  |  7715.68 |   0 |   955.268  |   0 |   0 |   348.307  |   8943.55  |  1249 |
| MADISON      |  5310.48 |  21363.7  |  9333.9  |   0 |  1296.86   |   0 |   0 |   304.178  |  10428.7   |  1359 |
| MONTGOMERY   |  4224.24 |  17548.9  |  8520.81 |   0 |   809.85   |   0 |   0 |   270.48   |   7947.72  |   914 |
| NASSAU       | 68231.8  | 184242    | 74665.1  |   0 | 16696.3    |   0 |   0 | 12285.3    |  80595.6   | 12633 |
| NIAGARA      | 17136    |  72834.5  | 34109    |   0 |  4462.23   |   0 |   0 |  1319.92   |  32943.3   |  4359 |
| ONEIDA       | 18991.9  |  75543.8  | 41987.4  |   0 |  3729.98   |   0 |   0 |  1324.74   |  28501.8   |  4650 |
| ORANGE       | 23023.7  |  99367.4  | 42151.8  |   0 |  8575.5    |   0 |   0 |  1356.45   |  47283.7   |  5008 |
| ORLEANS      |  3325.9  |  16141.6  |  7606.18 |   0 |   562.9    |   0 |   0 |   199.875  |   7772.69  |   922 |
| OSWEGO       | 10198.8  |  47198.6  | 18218.3  |   0 |  2404.22   |   0 |   0 |   814.309  |  25761.8   |  3125 |
| OTSEGO       |  4682.74 |  16551.4  |  9763.64 |   0 |   919.698  |   0 |   0 |   278.321  |   5589.72  |   874 |
| PUTNAM       |  6520.42 |  16400.9  |  7217.72 |   0 |  1063.21   |   0 |   0 |   650.82   |   7469.14  |  1090 |
| QUEENS       | 60849.4  | 301702    | 92023.6  |   0 | 51229.6    |   0 |   0 |  6647.66   | 151801     | 11484 |
| RENSSELAER   | 11003.7  |  47248.6  | 18841    |   0 |  3260.78   |   0 |   0 |   815.838  |  24331     |  2759 |
| RICHMOND     | 24312.1  |  87620    | 35615.1  |   0 | 12153      |   0 |   0 |  1609.15   |  38242.8   |  3880 |
| ROCKLAND     | 13156.5  |  42269.8  | 19481.4  |   0 |  3018.45   |   0 |   0 |  2073.22   |  17696.8   |  3164 |
| ST. LAWRENCE |  9019.14 |  37147.6  | 22143.5  |   0 |  2085.08   |   0 |   0 |   536.735  |  12382.2   |  2424 |
| SARATOGA     | 17334    |  50275.3  | 24252.3  |   0 |  4429.63   |   0 |   0 |   854.582  |  20738.8   |  3304 |
| SCHENECTADY  | 11417.3  |  40016.8  | 15846.8  |   0 |  5953.14   |   0 |   0 |   895.021  |  17321.9   |  2283 |
| SCHOHARIE    |  2867.81 |  10380.5  |  4127.53 |   0 |   513.531  |   0 |   0 |   146.138  |   5593.34  |   612 |
| SCHUYLER     |  1876.35 |   7081.26 |  3013.6  |   0 |   173.779  |   0 |   0 |    57.7755 |   3836.1   |   530 |
| SENECA       |  2660.28 |  14895.2  |  6410.4  |   0 |   363.688  |   0 |   0 |   189.322  |   7931.76  |   841 |
| SULLIVAN     |  5394.11 |  22813.2  |  9770.19 |   0 |   935.301  |   0 |   0 |   290.403  |  11817.3   |  1369 |
| TIOGA        |  4589.97 |  12859.5  |  7056.12 |   0 |   839.099  |   0 |   0 |   280.456  |   4683.8   |  1063 |
| TOMPKINS     |  5029.19 |  16578.3  |  7887.62 |   0 |  2161.76   |   0 |   0 |   507.387  |   6021.55  |  1066 |
| ULSTER       | 12948.9  |  45240.8  | 16973.1  |   0 |  4898.91   |   0 |   0 |  1076.65   |  22292.1   |  2824 |
| WARREN       |  6472.98 |  17891.5  |  9517.03 |   0 |   958.07   |   0 |   0 |   387.273  |   7029.13  |  1348 |
| WASHINGTON   |  5837.13 |  17037.1  |  8561.48 |   0 |   880.56   |   0 |   0 |   173.71   |   7421.33  |  1203 |
| WAYNE        |  7271.04 |  30427.8  | 11891.7  |   0 |  1704.42   |   0 |   0 |   332.732  |  16499     |  2029 |
| WYOMING      |  3435.97 |  13074.4  |  5840.14 |   0 |   507.659  |   0 |   0 |   181.087  |   6545.53  |   970 |
| YATES        |  2154.46 |  14066.4  |  5231.37 |   0 |   338.775  |   0 |   0 |   115.92   |   8380.31  |   731 |



labeled_formulas: G=B*E
H=B*G
G=C*E
H=C*G
G=D*E
H=D*G
G=E*F
G=E*H
G=E*I
G=E*J
G=E*K
H=F*G
H=G*I
H=G*J
H=G*K
G=(B+C)*E
H=(B+C)*G
G=(B+D)*E
H=(B+D)*G
G=(B*E)*F
G=(B*E)*H
G=(B*E)*I
G=(B*E)*J
G=(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)*G
G=(C*E)*F
G=(C*E)*H
G=(C*E)*I
G=(C*E)*J
G=(C*E)*K
H=(C+F)*G
H=(C*G)*I
H=(C*G)*J
H=(C*G)*K
G=(D*E)*F
G=(D*E)*H
G=(D*E)*I
G=(D*E)*J
G=(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
G=(E*F)*J
G=(E*F)*K
G=(E*H)*I
G=(E*H)*J
G=(E*H)*K
G=(E*I)*J
G=(E*I)*K
G=(E*J)*K
H=(F*G)*I
H=(F*G)*J
H=(F*G)*K
H=(G*I)*J
H=(G*I)*K
H=(G*J)*K
output: {"Arithmetic-Relationship": []}
source: $MMTU_HOME/resources/Relationship-finding/AR/table_119134___xlsx_41733d0294b45d708bccba64327d3a7c1798edcb.xlsx___line_198