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 |
|:----|:---------------|----------:|--------:|----------:|--------:|---------:|----:|----:|---------:|----------:|----------:|
| 1   | Adair          |   9325189 |       0 |    342994 |  923836 |        0 |   0 |   0 |  5428816 |  16020835 |  16020835 |
| 2   | Adams          |   2250493 |  128926 |    295102 |  445909 |        0 |   0 |   0 |      418 |   3120848 |   3120848 |
| 4   | Appanoose      |  10108150 |       0 |     60880 |  939120 |   514820 |   0 |   0 |   522160 |  12145130 |  12145130 |
| 5   | Audubon        |   3914410 |       0 |         0 |  500430 |   166210 |   0 |   0 |    45950 |   4627000 |   4627000 |
| 6   | Benton         |   3423000 |       0 |   1001000 |  178300 |        0 |   0 |   0 |        0 |   4602300 |   4602300 |
| 8   | Boone          |   1782497 |       0 |      9700 |  531899 |   605131 |   0 |   0 |   464812 |   3394039 |   3394039 |
| 9   | Bremer         |  24425620 |  282010 |  10203780 | 3591310 |        0 |   0 |   0 |  1269560 |  39772280 |  39772280 |
| 10  | Buchanan       |   7416328 |       0 |   3241252 | 2443112 |        0 |   0 |   0 |        0 |  13100692 |  13100692 |
| 11  | Buena Vista    |  19107790 |   33980 |   1315970 | 2712090 |        0 |   0 |   0 |   733940 |  23903770 |  23903770 |
| 12  | Butler         |  20113030 |       0 |         0 | 3909120 |   456190 |   0 |   0 |        0 |  24478340 |  24478340 |
| 14  | Carroll        |  14260690 |   33920 |   5186210 | 2771080 |        0 |   0 |   0 |   320550 |  22572450 |  22572450 |
| 17a | Mason City     |  36713120 |       0 |   9172190 | 2332980 |        0 |   0 |   0 |  1301950 |  49520240 |  49520240 |
| 18  | Cherokee       |  19062430 |       0 |    213780 | 1174669 |        0 |   0 |   0 |    50547 |  20501426 |  20501426 |
| 19  | Chickasaw      |   5897400 |  312500 |   1587600 | 1063000 |        0 |   0 |   0 |        0 |   8860500 |   8860500 |
| 20  | Clarke         |   4279910 |       0 |         0 |  377820 |        0 |   0 |   0 |        0 |   4657730 |   4657730 |
| 21  | Clay           |  36633870 | 5997570 |   2129070 | 1289310 |        0 |   0 |   0 |   262020 |  46311840 |  46311840 |
| 22  | Clayton        |   6846255 |       0 |    425152 | 3164589 |   789975 |   0 |   0 |    79986 |  11305957 |  11305957 |
| 23  | Clinton County |  11682276 |       0 |   1057223 | 2204791 |   388615 |   0 |   0 |        0 |  15332905 |  15332905 |
| 23a | Clinton City   |  22600210 |       0 |   1329724 | 2796266 |        0 |   0 |   0 |        0 |  26726200 |  26726200 |
| 24  | Crawford       |  15767416 |       0 |   5499210 | 1324484 |        0 |   0 |   0 |  1717580 |  24308690 |  24308690 |
| 27  | Decatur        |   2757385 |   42570 |         0 |  803817 |        0 |   0 |   0 |        0 |   3603772 |   3603772 |
| 28  | Delaware       |   9367570 |  349790 |    808920 | 1377500 |        0 |   0 |   0 |  1543110 |  13446890 |  13446890 |
| 29  | Des Moines     |  66402900 |       0 |   3710700 | 2155400 |        0 |   0 |   0 |  3426403 |  75695403 |  75695403 |
| 30  | Dickinson      |  35169100 |       0 |         0 | 1585700 | 77774900 |   0 |   0 |   171300 | 114701000 | 114701000 |
| 33  | Fayette        |  23316710 | 2100530 |   1128210 | 2449220 |        0 |   0 |   0 |    82400 |  29077070 |  29077070 |
| 35  | Franklin       |  12830800 |       0 |         0 |  651800 |        0 |   0 |   0 |   254100 |  13736700 |  13736700 |
| 36  | Fremont        |   3145130 |       0 |         0 |  640090 |        0 |   0 |   0 |    44650 |   3829870 |   3829870 |
| 37  | Greene         |   5180500 |  730200 |         0 |  324600 |        0 |   0 |   0 |    48700 |   6284000 |   6284000 |
| 38  | Grundy         |  15616221 |       0 |    840094 | 2035708 |        0 |   0 |   0 |   300324 |  18792347 |  18792347 |
| 39  | Guthrie        |   9005502 |       0 |         0 |  680814 |   486433 |   0 |   0 |        0 |  10172749 |  10172749 |
| 41  | Hancock        |  15170110 |       0 |    247590 | 1908890 |        0 |   0 |   0 |   195530 |  17522120 |  17522120 |
| 43  | Harrison       |   8834859 |       0 |         0 | 1601666 |        0 |   0 |   0 |    19047 |  10455572 |  10455572 |
| 44  | Henry          |   5851275 |       0 |         0 | 1133170 |        0 |   0 |   0 |   148884 |   7133329 |   7133329 |
| 45  | Howard         |   9716820 |  434300 |   3036010 | 1421100 |        0 |   0 |   0 |   284200 |  14892430 |  14892430 |
| 47  | Ida            |   3058418 |       0 |         0 |  838710 |        0 |   0 |   0 |    23964 |   3921092 |   3921092 |
| 50  | Jasper         |  28411980 |       0 |    828350 | 2248940 |        0 |   0 |   0 |   339500 |  31828770 |  31828770 |
| 51  | Jefferson      |  26890500 |  154000 |     89600 | 1670000 |        0 |   0 |   0 |   664200 |  29468300 |  29468300 |
| 52a | Iowa City      |  80638283 | 1489690 |  15909030 | 2990427 |        0 |   0 |   0 |  2383900 | 103411330 | 103411330 |
| 53  | Jones          |  30140150 |       0 |   3196740 | 2132070 |   610220 |   0 |   0 |        0 |  36079180 |  36079180 |
| 55  | Kossuth        |  17161642 |       0 |   7458725 | 1085041 |        0 |   0 |   0 |  2379895 |  28085303 |  28085303 |
| 56  | Lee            |  23715600 | 1437040 |   3109310 | 1291370 |        0 |   0 |   0 |  1059230 |  30612550 |  30612550 |
| 57a | Cedar Rapids   | 187676700 |       0 |  40628300 | 4173500 |        0 |   0 |   0 | 24445700 | 256924200 | 256924200 |
| 58  | Louisa         |   2081669 |       0 |         0 |  490183 |        0 |   0 |   0 |        0 |   2571852 |   2571852 |
| 59  | Lucas          |   3271105 |       0 |         0 |  391250 |   351145 |   0 |   0 |        0 |   4013500 |   4013500 |
| 60  | Lyon           |  35792792 |   94850 |   1565375 | 5105869 |        0 |   0 |   0 |        0 |  42558886 |  42558886 |
| 62  | Mahaska        |  30859260 | 2593400 |   1489810 | 2674470 |        0 |   0 |   0 |   423960 |  38040900 |  38040900 |
| 65  | Mills          |   8180116 |  150144 |         0 | 1081233 |        0 |   0 |   0 |   218515 |   9630008 |   9630008 |
| 67  | Monona         |   9933683 |       0 |    182586 | 1111861 |    37304 |   0 |   0 |        0 |  11265434 |  11265434 |
| 68  | Monroe         |   5816993 |  718119 |     86860 |  832540 |        0 |   0 |   0 |    12443 |   7466955 |   7466955 |
| 69  | Montgomery     |   5829430 |       0 |      6450 |  732860 |        0 |   0 |   0 |   177140 |   6745880 |   6745880 |
| 72  | Osceola        |   6860320 |   19800 |    279830 | 1127180 |        0 |   0 |   0 |    38970 |   8326100 |   8326100 |
| 73  | Page           |  15267952 |   57145 |     16860 | 1088350 |        0 |   0 |   0 |   229417 |  16659724 |  16659724 |
| 74  | Palo Alto      |  26135800 | 1031560 |   3435320 | 3023560 |        0 |   0 |   0 |        0 |  33626240 |  33626240 |
| 76  | Pocahontas     |   6591836 |       0 |    659634 |  499421 |        0 |   0 |   0 |    57639 |   7808530 |   7808530 |
| 78  | Pottawattamie  |  66400699 |  300800 |   6417500 | 6781602 |        0 |   0 |   0 |  8073146 |  87973747 |  87973747 |
| 80  | Ringgold       |   2766877 |       0 |         0 |  293379 |        0 |   0 |   0 |        0 |   3060256 |   3060256 |
| 82  | Scott          |  39308420 |       0 |   3708950 | 2525490 |     3940 |   0 |   0 |    96350 |  45643150 |  45643150 |
| 82a | Davenport      | 124331540 |       0 | 116857110 | 7223120 |        0 |   0 |   0 |  7904300 | 256316070 | 256316070 |
| 83  | Shelby         |   4144211 |  962885 |    810070 | 1052751 |        0 |   0 |   0 |    31894 |   7001811 |   7001811 |
| 85  | Story          |  24522700 |  102500 |    253300 | 1588300 |        0 |   0 |   0 |  3803900 |  30270700 |  30270700 |
| 85a | Ames           |  63723800 |       0 |   1956000 | 2360900 |        0 |   0 |   0 |  7319400 |  75360100 |  75360100 |
| 86  | Tama           |   8371230 |       0 |    259470 | 1016530 |  1244250 |   0 |   0 |   484260 |  11375740 |  11375740 |
| 87  | Taylor         |   2939612 |       0 |      6150 |  728270 |        0 |   0 |   0 |   177520 |   3851552 |   3851552 |
| 88  | Union          |    586160 |       0 |    279220 |  285470 |        0 |   0 |   0 |        0 |   1150850 |   1150850 |
| 89  | Vn Buren       |   3718970 |       0 |    298910 |  638160 |   679890 |   0 |   0 |        0 |   5335930 |   5335930 |
| 90  | Wapello        |  18976286 |       0 |   1893290 | 1730828 |  1968380 |   0 |   0 |   404505 |  24973289 |  24973289 |
| 91  | Warren         |  46619200 |       0 |    396300 | 2379900 |  2544700 |   0 |   0 |        0 |  51940100 |  51940100 |
| 92  | Washington     |  14403100 |       0 |    117100 | 1190400 |  5179100 |   0 |   0 |  1973500 |  22863200 |  22863200 |
| 93  | Wayne          |   5412290 |       0 |         0 |  649480 |        0 |   0 |   0 |        0 |   6061770 |   6061770 |
| 95  | Winnebago      |   6307450 |       0 |         0 |  978650 |        0 |   0 |   0 |   568795 |   7854895 |   7854895 |
| 96  | Winneshiek     |   7507240 |       0 |    132020 |  408110 |        0 |   0 |   0 |        0 |   8047370 |   8047370 |
| 97  | Woodbury       |  12842530 |       0 |    574670 | 2637130 |        0 |   0 |   0 |  1125430 |  17179760 |  17179760 |
| 98  | Worth          |   7005895 |       0 |         0 |  974204 |        0 |   0 |   0 |        0 |   7980099 |   7980099 |
| 99  | Wright         |  19017400 |       0 |    728700 | 1533700 |        0 |   0 |   0 |        0 |  21279800 |  21279800 |



labeled_formulas: I=C*H
I=D*H
I=E*H
I=F*H
I=G*H
I=H*J
I=H*K
I=H*L
L=H+K
L=I+K
I=(C*D)*H
I=(C*E)*H
I=(C+F)*H
I=(C+G)*H
I=(C*H)*J
I=(C*H)*K
I=(C*H)*L
L=(H/C)+K
I=(D+E)*H
I=(D*F)*H
I=(D*G)*H
L=(D*G)+K
I=(D*H)*J
I=(D*H)*K
I=(D*H)*L
L=(D*H)+K
L=(D*I)+K
I=(E*F)*H
I=(E*G)*H
I=(E*H)*J
I=(E*H)*K
I=(E*H)*L
L=(E*H)+K
L=(E*I)+K
I=(F+G)*H
I=(F*H)*J
I=(F*H)*K
I=(F*H)*L
L=(H/F)+K
I=(G*H)*J
I=(G*H)*K
I=(G*H)*L
L=(G*H)+K
L=(G*I)+K
I=(H*J)*K
I=(H*J)*L
L=(H*I)+K
L=(H*J)+K
L=(I*J)+K
output: {"Arithmetic-Relationship": []}
source: $MMTU_HOME/resources/Relationship-finding/AR/table_521713___xlsx_fe4a9e9ec6c7d904e6fb34227388286f801b9cd9.xlsx___line_942