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 |
|:----------------|-----------:|-----------:|-----------:|----:|-----------:|----:|----:|------------:|-----------:|------:|
| ALPINE          |    121.643 |    246.761 |    105.913 |   0 |    21.2244 |   0 |   0 |     6.16589 |    113.458 |    16 |
| AMADOR          |   4471.05  |  14020.8   |   7029.16  |   0 |   465.393  |   0 |   0 |   240.362   |   6285.88  |   723 |
| BUTTE           |  18815     |  81920.7   |  46026.9   |   0 |  7824.38   |   0 |   0 |  1274.39    |  26795     |  4940 |
| CALAVERAS       |   5776.69  |  19895.1   |   9363.1   |   0 |   622.829  |   0 |   0 |   575.668   |   9333.51  |  1391 |
| COLUSA          |   1403.95  |   3798.17  |   1962.71  |   0 |   150.243  |   0 |   0 |    97.7854  |   1587.43  |   214 |
| DEL NORTE       |   3092.62  |  16018.9   |  10953.5   |   0 |   416.364  |   0 |   0 |   122.434   |   4526.56  |   874 |
| EL DORADO       |  15624.6   |  56097.2   |  30903.8   |   0 |  3123.92   |   0 |   0 |  1462.5     |  20607     |  2818 |
| GLENN           |   1701.96  |   6123.77  |   3580.03  |   0 |   401.008  |   0 |   0 |   155.694   |   1987.04  |   440 |
| HUMBOLDT        |  11501.2   |  58787.1   |  30433.4   |   0 |  2938.93   |   0 |   0 |   625.869   |  24788.9   |  4643 |
| IMPERIAL        |   6933.27  |  30555.9   |  14018     |   0 |  3848.27   |   0 |   0 |   208.468   |  12481.2   |  1672 |
| INYO            |   1542.72  |   5301.25  |   2618.34  |   0 |   109.366  |   0 |   0 |   130.978   |   2442.57  |   325 |
| KERN            |  46451.1   | 160298     |  87819.3   |   0 | 17059.6    |   0 |   0 |  2896.86    |  52522.6   |  8013 |
| KINGS           |  10542.8   |  44666.5   |  24100.9   |   0 |  7642.96   |   0 |   0 |  1058.37    |  11864.3   |  1870 |
| LAKE            |   7544.79  |  37904.7   |  19024.2   |   0 |   750.691  |   0 |   0 |   349.971   |  17779.7   |  1932 |
| LASSEN          |   3774.97  |  12691.9   |   6515.55  |   0 |   692.733  |   0 |   0 |    76.2668  |   5407.31  |   739 |
| MADERA          |   9223.18  |  38660.7   |  17526.9   |   0 |  1755.6    |   0 |   0 |   538.446   |  18839.7   |  2055 |
| MARIN           |  15721.6   |  46637.3   |  18697.5   |   0 |  2583.53   |   0 |   0 |  2540.46    |  22815.9   |  2030 |
| MARIPOSA        |   2366.5   |   9333.48  |   4934.84  |   0 |   207.344  |   0 |   0 |   127.149   |   4064.15  |   560 |
| MENDOCINO       |   7531.47  |  45317.2   |  22922.5   |   0 |   875.658  |   0 |   0 |   479.121   |  21039.9   |  2295 |
| MODOC           |    836.686 |   5461     |   2899.85  |   0 |   115.508  |   0 |   0 |    93.5167  |   2352.12  |   363 |
| MONO            |    832.296 |   2139.24  |    996.257 |   0 |    91.0601 |   0 |   0 |    35.4781  |   1016.44  |   149 |
| MONTEREY        |  22703.3   | 128060     |  69958.7   |   0 |  8845.02   |   0 |   0 |  2622.76    |  46633.9   |  6050 |
| NAPA            |   9071.01  |  33404.7   |  20134.9   |   0 |  1612.76   |   0 |   0 |   791.121   |  10865.9   |  1799 |
| NEVADA          |  11285.3   |  36576     |  20931.1   |   0 |  1861.95   |   0 |   0 |   845.259   |  12937.7   |  2273 |
| ORANGE          | 139719     | 423976     | 186104     |   0 | 64362.9    |   0 |   0 | 14608.4     | 158901     | 20437 |
| PLACER          |  32493.2   |  94716.5   |  51593.8   |   0 |  9760.43   |   0 |   0 |  2454.38    |  30908     |  4918 |
| PLUMAS          |   2414.28  |  11191.8   |   5499.32  |   0 |   309.255  |   0 |   0 |   173.783   |   5209.45  |   685 |
| SACRAMENTO      |  98675.3   | 424894     | 211600     |   0 | 42351.9    |   0 |   0 |  7602.15    | 163340     | 18935 |
| SAN BENITO      |   3038.97  |  13273.2   |   6297.41  |   0 |   733.672  |   0 |   0 |    91.8669  |   6150.3   |   717 |
| SAN JOAQUIN     |  37435.9   | 155429     |  72092     |   0 | 10420.2    |   0 |   0 |  2157.65    |  70759     |  7886 |
| SAN LUIS OBISPO |  22625.6   |  59771.1   |  34642.3   |   0 |  5426.74   |   0 |   0 |  2230.06    |  17472     |  4357 |
| SANTA BARBARA   |  26607.8   |  84192.3   |  44365.5   |   0 |  9803.64   |   0 |   0 |  3506.93    |  26516.2   |  4877 |
| SANTA CRUZ      |  14042.3   |  59692.4   |  26831.9   |   0 |  3273.85   |   0 |   0 |  1217.46    |  28369.2   |  3194 |
| SHASTA          |  18107.2   | 104817     |  59382.3   |   0 |  4478.1    |   0 |   0 |  1219.91    |  39736.5   |  6249 |
| SIERRA          |    357.222 |   1817.11  |    917.891 |   0 |    72.4474 |   0 |   0 |     6.4154  |    820.355 |   106 |
| SISKIYOU        |   4611.1   |  18329.6   |  10880.1   |   0 |   561.082  |   0 |   0 |   406.543   |   6481.94  |  1462 |
| SOLANO          |  39655.4   | 201540     | 121874     |   0 | 20879.8    |   0 |   0 |  3262.43    |  55523.9   |  7475 |
| SONOMA          |  33245.2   | 123399     |  54939.5   |   0 |  8589.43   |   0 |   0 |  4001.22    |  55868.8   |  5968 |
| STANISLAUS      |  27018.4   | 120776     |  59237.1   |   0 |  8606.87   |   0 |   0 |  1823.91    |  51108.3   |  6128 |
| SUTTER          |   6962.74  |  30583.7   |  20280.5   |   0 |  2907.44   |   0 |   0 |   596.62    |   6799.18  |  1279 |
| TEHAMA          |   6410.53  |  29571.8   |  15476.5   |   0 |  1135.05   |   0 |   0 |   228.831   |  12731.5   |  1901 |
| TRINITY         |   1550.21  |   8394.34  |   4870.09  |   0 |   140.725  |   0 |   0 |   161.351   |   3222.18  |   527 |
| TULARE          |  18403.7   |  87411.8   |  46989.5   |   0 |  6675.3    |   0 |   0 |   940.003   |  32807     |  5145 |
| TUOLUMNE        |   7182.69  |  30804.2   |  14875     |   0 |   838.861  |   0 |   0 |   630.327   |  14460     |  2140 |
| VENTURA         |  51789     | 147295     |  71136.8   |   0 | 22144.4    |   0 |   0 |  4543.07    |  49470.4   |  7673 |
| YOLO            |   8533.4   |  43217.1   |  22226.6   |   0 |  6567.66   |   0 |   0 |   901.978   |  13520.9   |  1852 |
| YUBA            |   5303.04  |  32087.2   |  18968     |   0 |  4307.01   |   0 |   0 |   214.79    |   8597.37  |  1320 |



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_119104___xlsx_41733d0294b45d708bccba64327d3a7c1798edcb.xlsx___line_168