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 |          L |          M |
|:----|:---------------|----------:|----------:|---------:|--------:|----------:|--------:|------:|----:|----:|-----------:|-----------:|
| 1   | ADAIR          |   3841074 |    703449 |  1851304 |   83743 |   1102494 |       0 |     0 |   0 |   0 |    7582064 |    7582064 |
| 2   | ADAMS          |   2684371 |    147739 |  1320457 |   58136 |   3131007 |       0 |     0 |   0 |   0 |    7341710 |    7341710 |
| 3   | ALLAMAKEE      |  15213380 |    234400 |  3221500 |   76100 |  11099600 |       0 |     0 |   0 |   0 |   29844980 |   29844980 |
| 4   | APPANOOSE      |  12253250 |    789420 |  8285030 |  142940 |  13817760 |       0 |     0 |   0 |   0 |   35288400 |   35288400 |
| 5   | AUDUBON        |   4653390 |    140190 |   916400 |   64760 |   3788500 |       0 |     0 |   0 |   0 |    9563240 |    9563240 |
| 8   | BOONE          |   2078355 |    276762 |  1415839 |   95800 |  13322197 |       0 |     0 |   0 |   0 |   17188953 |   17188953 |
| 9   | BREMER         |  37497440 |  98373070 |  5441850 |  655280 |  34330820 |       0 |     0 |   0 |   0 |  176298460 |  176298460 |
| 10  | BUCHANAN       |  13005230 |    164382 |  6383111 |  688350 |   5546180 |       0 |     0 |   0 |   0 |   25787253 |   25787253 |
| 11  | BUENA VISTA    |  24092640 |  17024070 |  1476970 |  250430 |   7876065 |       0 |     0 |   0 |   0 |   50720175 |   50720175 |
| 12  | BUTLER         |  11636082 |    692892 |  3120933 |  242924 |    199608 |       0 |     0 |   0 |   0 |   15892439 |   15892439 |
| 13  | CALHOUN        |  16207390 |   1113210 |  4256120 |  210100 |   3276874 |       0 |     0 |   0 |   0 |   25063694 |   25063694 |
| 14  | CARROLL        |  21069850 |         0 |  1154900 |  863130 |  31083940 |       0 |     0 |   0 |   0 |   54171820 |   54171820 |
| 15  | CASS           |  13752920 |   1574310 |        0 |   86880 |  10429100 |       0 |     0 |   0 |   0 |   25843210 |   25843210 |
| 16  | CEDAR          |  11210980 |   1204170 |        0 |  349780 |   8410350 |       0 |     0 |   0 |   0 |   21175280 |   21175280 |
| 18  | CHEROKEE       |  20145759 |    344454 |   975300 |   62446 |  16769670 |       0 |     0 |   0 |   0 |   38297629 |   38297629 |
| 19  | CHICKASAW      |   8772800 |         0 |   564500 |   61900 |   5905500 |       0 |     0 |   0 |   0 |   15304700 |   15304700 |
| 21  | CLAY           |  41624340 |    846550 |  1596720 |  314610 |  12598900 |       0 |     0 |   0 |   0 |   56981120 |   56981120 |
| 22  | CLAYTON        |  10061620 |    686980 |        0 |  122255 |  23329601 |       0 |     0 |   0 |   0 |   34200456 |   34200456 |
| 23  | CLINTON COUNTY |  15765064 |   1130487 |  2274806 |  257580 |   3636088 |       0 |     0 |   0 |   0 |   23064025 |   23064025 |
| 25  | DALLAS         | 140009530 |   2394380 |  7985360 | 2447310 | 138208540 |       0 |     0 |   0 |   0 |  291045120 |  291045120 |
| 26  | DAVIS          |   7783130 |         0 |  2379370 |   34430 |   1891010 |       0 |     0 |   0 |   0 |   12087940 |   12087940 |
| 28  | DELAWARE       |  10937900 |    583700 |  2014000 |   73000 |  33871800 |       0 |     0 |   0 |   0 |   47480400 |   47480400 |
| 31  | DUBUQUE COUNTY |  39508263 |   3731380 |  2980770 |  326200 |  15121090 |       0 |     0 |   0 |   0 |   61667703 |   61667703 |
| 32  | EMMET          |  14546300 |    415500 |  1417800 |  170400 |  15874700 |       0 |     0 |   0 |   0 |   32424700 |   32424700 |
| 34  | FLOYD          |  25066350 |   1074700 |   450000 |   74340 |   8091820 |       0 |     0 |   0 |   0 |   34757210 |   34757210 |
| 35  | FRANKLIN       |  13922800 |    173100 |  2963600 |   71000 |   5612500 |       0 |     0 |   0 |   0 |   22743000 |   22743000 |
| 36  | FREMONT        |   3758700 |    618700 |  1955060 |  122650 |   2659830 |       0 | 55300 |   0 |   0 |    9170240 |    9170240 |
| 37  | GREENE         |   6637000 |    267000 |  1925500 |   63200 |    942300 |       0 |     0 |   0 |   0 |    9835000 |    9835000 |
| 38  | GRUNDY         |  15699947 |         0 |  1642635 |    2558 |    488287 |       0 |     0 |   0 |   0 |   17833427 |   17833427 |
| 40  | HAMILTON       |  25247240 |   1899610 |  1169420 |   67980 |   2868110 |       0 |     0 |   0 |   0 |   31252360 |   31252360 |
| 41  | HANCOCK        |  15484130 |    694570 |   166890 |  160270 |   4186960 |       0 |     0 |   0 |   0 |   20692820 |   20692820 |
| 42  | HARDIN         |  51197099 |    450857 |  3775320 |  550160 |  14365984 |       0 |     0 |   0 |   0 |   70339420 |   70339420 |
| 43  | HARRISON       |   7981085 |     77808 |  2845322 |   73491 |   5116760 |       0 |     0 |   0 |   0 |   16094466 |   16094466 |
| 44  | HENRY          |   7153969 |  10284777 |  1279440 |   91294 |   1757079 |       0 |     0 |   0 |   0 |   20566559 |   20566559 |
| 46  | HUMBOLDT       |   3543605 |         0 |  2120702 |   83140 |   5490825 |       0 |     0 |   0 |   0 |   11238272 |   11238272 |
| 47  | IDA            |   4175814 |     31140 |  1100330 |   71339 |   1493786 |       0 |     0 |   0 |   0 |    6872409 |    6872409 |
| 48  | IOWA           |  14799020 |    690710 |  1945190 |  283040 |  28655600 |       0 |     0 |   0 |   0 |   46373560 |   46373560 |
| 49  | JACKSON        |   8625660 |         0 |        0 |       0 |   7188500 |       0 |     0 |   0 |   0 |   15814160 |   15814160 |
| 52  | JOHNSON        |  19460234 |    352238 |   169490 | 1080558 |   2519652 |       0 |     0 |   0 |   0 |   23582172 |   23582172 |
| 52a | IOWA CITY      |  89282500 |   6024640 |        0 |  564960 | 158077941 |       0 |     0 |   0 |   0 |  253950041 |  253950041 |
| 53  | JONES          |  36585380 |   8206930 |  7156570 |  243950 |  13892010 |       0 |     0 |   0 |   0 |   66084840 |   66084840 |
| 55  | KOSSUTH        |  23158668 |    184932 |  2273929 |  288012 |  18110828 |       0 |     0 |   0 |   0 |   44016369 |   44016369 |
| 56  | LEE            |  30149650 |    762040 |  8783170 |  334310 |  14506070 |       0 |     0 |   0 |   0 |   54535240 |   54535240 |
| 57  | LINN           |  94006100 |   5721425 |  4144900 | 2076600 |  37346200 |       0 |     0 |   0 |   0 |  143295225 |  143295225 |
| 58  | LOUISA         |   2580270 |    322213 |        0 |  196164 |   1072623 |       0 |     0 |   0 |   0 |    4171270 |    4171270 |
| 60  | LYON           |  35825107 |    258080 |  2717020 |   34270 |   7540847 |       0 |     0 |   0 |   0 |   46375324 |   46375324 |
| 61  | MADISON        |  19664700 |   1439400 |  3891900 |   94700 |   1321200 |       0 |     0 |   0 |   0 |   26411900 |   26411900 |
| 62  | MAHASKA        |  39631090 |  16338910 |  3300830 |  168670 |   4577770 |       0 |     0 |   0 |   0 |   64017270 |   64017270 |
| 63  | MARION         |  95491400 |  65857360 |  3694290 |  111600 |  86912490 |       0 |     0 |   0 |   0 |  252067140 |  252067140 |
| 64  | MARSHALL       |  21615530 |    269216 |  2402236 |  204416 |  17967436 |       0 |     0 |   0 |   0 |   42458834 |   42458834 |
| 65  | MILLS          |   9560150 |    612207 |  2108592 |  232946 |   2771357 |       0 |     0 |   0 |   0 |   15285252 |   15285252 |
| 66  | MITCHELL       |  10509815 |     96690 |  2034140 |  457435 |   2899878 |       0 |     0 |   0 |   0 |   15997958 |   15997958 |
| 67  | MONONA         |  11156766 |    637786 |  1609666 |  251579 |  14712294 |       0 |     0 |   0 |   0 |   28368091 |   28368091 |
| 69  | MONTGOMERY     |   6088730 |   1759590 |  3206470 |   73170 |  16462780 |       0 |     0 |   0 |   0 |   27590740 |   27590740 |
| 70  | MUSCATINE      |  62143784 |         0 |  7664330 |  685510 |  60962262 |       0 |     0 |   0 |   0 |  131455886 |  131455886 |
| 71  | O'BRIEN        |  25350310 |         0 |   598540 |  483990 |   4243880 |       0 |     0 |   0 |   0 |   30676720 |   30676720 |
| 72  | OSCEOLA        |   8388426 |     31280 |        0 |  245980 |   5984174 |       0 |     0 |   0 |   0 |   14649860 |   14649860 |
| 73  | PAGE           |  16729714 |    805980 |  5102385 |  580460 |  15619813 |       0 |     0 |   0 |   0 |   38838352 |   38838352 |
| 74  | PALO ALTO      |  34170360 |     67350 |  2563990 |  995940 |   9403980 |       0 |     0 |   0 |   0 |   47201620 |   47201620 |
| 75  | PLYMOUTH       |  19840010 |    415120 |  3032130 |   68500 |   5901130 |       0 |     0 |   0 |   0 |   29256890 |   29256890 |
| 76  | POCAHONTAS     |   8455041 |         0 |   427976 |  159669 |   1969493 |       0 |     0 |   0 |   0 |   11012179 |   11012179 |
| 77  | POLK           | 763889750 | 413970160 | 55836790 | 2205280 | 943906960 |       0 |     0 |   0 |   0 | 2179808940 | 2179808940 |
| 79  | POWESHIEK      |  11268650 | 219344270 |        0 |   48290 |  32309470 |       0 |     0 |   0 |   0 |  262970680 |  262970680 |
| 80  | RINGGOLD       |   2205903 |    185537 |  1027741 |   98720 |     89101 |       0 |     0 |   0 |   0 |    3607002 |    3607002 |
| 81  | SAC            |  14963940 |    114130 |   619300 |  111790 |   3596470 |       0 |     0 |   0 |   0 |   19405630 |   19405630 |
| 82a | DAVENPORT      | 256512400 |  76650400 | 10627170 |  895940 | 305912054 |       0 |     0 |   0 |   0 |  650597964 |  650597964 |
| 83  | SHELBY         |   6771405 |         0 |  1067536 |   12806 |   6997998 |       0 |     0 |   0 |   0 |   14849745 |   14849745 |
| 84  | SIOUX COUNTY   | 174998960 | 170882450 |  5549980 |  181340 |  55172580 |       0 |     0 |   0 |   0 |  406785310 |  406785310 |
| 85  | STORY          |  30347900 |    249400 |  1113900 |  535800 |  34788300 |       0 |     0 |   0 |   0 |   67035300 |   67035300 |
| 85a | AMES           |  68785880 |    888600 |  5523700 |  321900 |  47662600 |       0 |     0 |   0 |   0 |  123182680 |  123182680 |
| 86  | TAMA           |  13834060 |         0 |  1364170 |  433020 |   2554590 |       0 |     0 |   0 |   0 |   18185840 |   18185840 |
| 87  | TAYLOR         |   3986352 |     98800 |  1295526 |  161770 |    321870 |       0 |     0 |   0 |   0 |    5864318 |    5864318 |
| 88  | UNION          |   1146630 |         0 |        0 |   15000 |    125000 |       0 |     0 |   0 |   0 |    1286630 |    1286630 |
| 89  | VAN BUREN      |   4734250 |    362370 |   186890 |  231880 |  15715060 |       0 |     0 |   0 |   0 |   21230450 |   21230450 |
| 90  | WAPELLO        |  25807570 |    719581 |  2805359 |  292276 |  13754722 |       0 |     0 |   0 |   0 |   43379508 |   43379508 |
| 92  | WASHINGTON     |  21796786 |    161800 |  1836100 |  167200 |  15952900 |       0 |     0 |   0 |   0 |   39914786 |   39914786 |
| 93  | WAYNE          |   4076490 |   1137723 |  2358709 |  123933 |     97543 |       0 |     0 |   0 |   0 |    7794398 |    7794398 |
| 95  | WINNEBAGO      |   7854895 |         0 |  1087300 |   41420 |   1028565 |       0 |     0 |   0 |   0 |   10012180 |   10012180 |
| 96  | WINNESHIEK     |   8229670 |  69225550 |  4655800 |   96220 |   9489950 |       0 |     0 |   0 |   0 |   91697190 |   91697190 |
| 97a | SIOUX CITY     | 144418400 |  96450400 |        0 |  201900 | 281439100 | 6006100 |     0 |   0 |   0 |  528515900 |  528515900 |
| 98  | WORTH          |   7993671 |         0 |  2034731 |  105255 |   3978953 |       0 |     0 |   0 |   0 |   14112610 |   14112610 |
| 99  | WRIGHT         |  13018800 |    549400 |  1979800 |   90800 |   2935600 |       0 |     0 |   0 |   0 |   18574400 |   18574400 |



labeled_formulas: M=(H/C)+L
J=(D*E)*H
K=(D*E)*H
M=(H/G)+L
output: {"Arithmetic-Relationship": []}
source: $MMTU_HOME/resources/Relationship-finding/AR/table_450744___xlsx_daa4317c9627f49e50b0d26b2cd97bde17ca05f3.xlsx___line_957