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 |      N |   O |   P |
|:---------------|---------:|----------:|---------:|----------:|---------:|----------:|---------:|----------:|----------:|-------:|-------:|----------:|-------:|----:|----:|
| Andrew         |  13538   |   4800    |  13538   |   1200    |   6769   |   1200    |  33845   |   7200    | 0.212735  |   4738 |   3790 |  20000    |   3790 |   0 |   1 |
| Atchison       |      0   |      0    |      0   |      0    |      0   |      0    |      0   |      0    | 0         |      0 |      0 |      0    |      0 |   0 |   0 |
| Audrain        |   9500   |      0    |   9500   |   4575    |   4750   |      0    |  23750   |   4575    | 0.192632  |   3325 |   2660 |  10000    |   2660 |   0 |   1 |
| Barry          |  13538   |      0    |  34570.6 |  41349.1  |   6769   |  46770.9  |  54877.6 |  88120    | 1.60576   |   5077 |   6092 |  40000    |   6092 |   0 |   1 |
| Barton         |  13538   |  14708.1  |  13538   |  23346.8  |   6769   |   7934.47 |  33845   |  45989.4  | 1.35882   |   5754 |   6905 |  94500    |   6905 |   0 |   1 |
| Bates          |  10450   |  12274.4  |  10450   |      0    |   5225   |      0    |  26125   |  12274.4  | 0.469834  |   3658 |   2926 | 237487    |   2926 |   0 |   1 |
| Benton         |  16116   |   9359.21 |  20951   |  32570.5  |  24094   |      0    |  61161   |  41929.7  | 0.685562  |  20480 |  18432 |  61800    |  18432 |   0 |   1 |
| Bollinger      |  13876   |   2734.02 |  13876   |   8314.28 |  13876   |    400    |  41628   |  11448.3  | 0.275014  |  10407 |   8326 | 111600    |   8326 |   0 |   1 |
| Boone          |  13538   |  10082.6  |  13538   |  19166    |   6769   |  26509.6  |  33845   |  55758.3  | 1.64746   |   5077 |   6092 | 137725    |   6092 |   0 |   1 |
| Buchanan       |  13538   |      0    |  13538   |      0    |   2000   |      0    |  29076   |      0    | 0         |   2000 |   2000 |  20000    |   2000 |   1 |   1 |
| Butler         |  13538   |  12706.5  |  13538   |   9615.37 |   6769   |      0    |  33845   |  22321.9  | 0.659532  |   5077 |   4569 |  15000    |   4569 |   0 |   1 |
| Caldwell       |  20858   |  23734.1  |  20858   |  26059.7  |  20858   |  58138.2  |  62574   | 107932    | 1.72487   |  17729 |  21275 |  94000    |  21275 |   0 |   1 |
| Callaway       |  13538   |      0    |  13538   |      0    |   2000   |      0    |  29076   |      0    | 0         |   2000 |   2000 | 170000    |   2000 |   1 |   1 |
| Camden         |  15337   |  10541    |  15337   |  23341.6  |  15337   |  11479.8  |  46011   |  45362.3  | 0.985902  |  13036 |  15643 |  34300    |  15643 |   0 |   1 |
| Cape Girardeau |  15711   |   2581.57 |  20424   |   4642.06 |  23488   |      0    |  59623   |   7223.63 | 0.121155  |  16442 |  13154 |  75000    |  13154 |   0 |   1 |
| Carroll        |  13538   |      0    |  13538   |   3590.67 |   6769   |      0    |  33845   |   3590.67 | 0.106092  |   4738 |   3790 |  23700    |   3790 |   0 |   1 |
| Carter         |  13876   |      0    |  13876   |  60598.3  |   6938   |   3177.73 |  34690   |  63776.1  | 1.83846   |   5897 |   7076 |  65533.8  |   7076 |   0 |   1 |
| Cass           |  16951   |      0    |  16951   |  17256.6  |   8476   |  10719.6  |  42378   |  27976.2  | 0.660158  |   5933 |   5340 |  55390    |   5340 |   0 |   1 |
| Cedar          |   9500   |    400    |   9500   |      0    |   4750   |      0    |  23750   |    400    | 0.0168421 |   3325 |   2660 |  17200    |   2660 |   0 |   1 |
| Chariton       |   9386   |      0    |   9386   |      0    |   4693   |      0    |  23465   |      0    | 0         |   2000 |   2000 |   4200    |   2000 |   1 |   1 |
| Christian      |  13538   |  19112.9  |  13538   |  31803.3  |  13538   |  44477.6  |  40614   |  95393.8  | 2.34879   |  11507 |  13808 | 100300    |  13808 |   0 |   1 |
| Clay           |      0   |      0    |      0   |      0    |   5000   |      0    |   5000   |      0    | 0         |   2000 |      0 |      0    |      0 |   0 |   0 |
| Cole           |  27563   |  13284.3  |  27563   |  18369.8  |  27563   |   2169.29 |  82689   |  33823.4  | 0.409044  |  19294 |  15435 |  85268    |  15435 |   0 |   1 |
| Cooper         |  13538   |  48174.5  |  13538   |   8284.5  |  15569   |    400    |  42645   |  56859    | 1.33331   |  13234 |  15881 |  73200    |  15881 |   0 |   1 |
| Crawford       |  17599.9 |  11129.4  |  19452   |  40267.1  |  22370   |  35772.2  |  59421.9 |  87168.7  | 1.46695   |  19015 |  22818 |  50000    |  22818 |   0 |   1 |
| Dade           |  17949   |      0    |  17949   |  55428.3  |   8975   |      0    |  44873   |  55428.3  | 1.23523   |   7629 |   9155 |  20800    |   9155 |   0 |   1 |
| Dallas         |  26122   |  24315.1  |  33959   |  22635.5  |  39053   |  35277.2  |  99134   |  82227.8  | 0.829461  |  33195 |  29876 | 120000    |  29876 |   0 |   1 |
| Dekalb         |  13538   |      0    |  13538   |      0    |   6769   |      0    |  33845   |      0    | 0         |   4738 |   3790 |  24000    |   3790 |   0 |   1 |
| Dent           |  31240   |  86897.4  |  40612   |  76424.2  |  46704   | 130551    | 118556   | 293873    | 2.47877   |  39698 |  47638 | 358800    |  47638 |   0 |   1 |
| Dunklin        |  13538   |      0    |  13538   |      0    |   2000   |      0    |  29076   |      0    | 0         |   2000 |   2000 |  20500    |   2000 |   1 |   1 |
| Franklin       |  25398   |  29218.5  |  33017   |    400    |  37970   |  49323.9  |  96385   |  78942.4  | 0.819033  |  32275 |  29048 | 112000    |  29048 |   0 |   1 |
| Gasconade      |  13538   |  32460.3  |  13538   |  13275.2  |  15569   |      0    |  42645   |  45735.5  | 1.07247   |  13234 |  15881 |  61800    |  15881 |   0 |   1 |
| Gentry         |  13538   |      0    |  13538   |      0    |   6769   |      0    |  33845   |      0    | 0         |   2000 |   2000 |  26600    |   2000 |   1 |   1 |
| Greene         |  32977   |   8863.84 |  32977   |  12050    |  32977   |  28191.9  |  98931   |  49105.7  | 0.496363  |  24733 |  19786 | 120000    |  19786 |   0 |   1 |
| Grundy         |   1805   |      0    |   1805   |      0    |    903   |      0    |   4513   |      0    | 0         |   2000 |   2000 |  12000    |   2000 |   1 |   1 |
| Harrison       |  21511   |   3200    |  21511   |  20292.9  |  10756   |  16812.3  |  53778   |  40305.2  | 0.749474  |   8067 |   7260 | 108000    |   7260 |   0 |   1 |
| Henry          |  18191   |  16228    |  23648   |      0    |  23648   |    800    |  65487   |  17028    | 0.260021  |  17736 |  14189 |  59500    |  14189 |   0 |   1 |
| Hickory        |  34914   | 106859    |  45388   |   3935.67 |  52196   | 110238    | 132498   | 221033    | 1.6682    |  44367 |  53240 | 312961    |  53240 |   0 |   1 |
| Holt           |      0   |      0    |      0   |      0    |      0   |      0    |      0   |      0    | 0         |      0 |      0 |      0    |      0 |   0 |   0 |
| Howard         |   4000   |      0    |   4000   |      0    |   2000   |      0    |  10000   |      0    | 0         |   2000 |   2000 |   4000    |   2000 |   1 |   1 |
| Howell         |  19984   | 100309    |  25979   |  80369.4  |  29876   |   1030.92 |  75839   | 181709    | 2.39599   |  25395 |  30474 | 103300    |  30474 |   0 |   1 |
| Iron           |  13876   |      0    |  13876   |  17321    |   6938   |      0    |  34690   |  17321    | 0.499307  |   4857 |   3886 | 100000    |   3886 |   0 |   1 |
| Jackson        |      0   |      0    |      0   |      0    |      0   |      0    |      0   |      0    | 0         |      0 |      0 |      0    |      0 |   0 |   0 |
| Jasper         |  14963   |      0    |  30526.9 |  11074.9  |  40230.1 |  27463.8  |  85720   |  38538.7  | 0.449588  |  16778 |  13422 | 110000    |  13422 |   0 |   1 |
| Jefferson      |  15337   |      0    |  19938   |   9028.23 |  22929   |      0    |  58204   |   9028.23 | 0.155114  |  19490 |  15592 | 108226    |  15592 |   0 |   1 |
| Johnson        |  13538   |      0    |  13538   |      0    |   6769   |      0    |  33845   |      0    | 0         |   4738 |   3790 |  20000    |   3790 |   0 |   1 |
| Knox           |  13876   |      0    |  13876   |      0    |   6938   |      0    |  34690   |      0    | 0         |   4857 |   3886 |  42461    |   3886 |   0 |   1 |
| Laclede        |  14963   |  13751.2  |  14963   |      0    |   7482   |      0    |  37408   |  13751.2  | 0.367599  |   5237 |   4190 |  25400    |   4190 |   0 |   1 |
| Lafayette      |      0   |      0    |      0   |      0    |      0   |      0    |      0   |      0    | 0         |      0 |      0 |      0    |      0 |   0 |   0 |
| Lawrence       |  14963   |    400    |  19452   |    800    |  22370   |    400    |  56785   |   1600    | 0.0281765 |  15659 |  12527 | 169800    |  12527 |   0 |   1 |
| Lewis          |      0   |      0    |  10000   |      0    |   2000   |      0    |  12000   |      0    | 0         |      0 |      0 |      0    |      0 |   0 |   0 |
| Lincoln        |   3339   |      0    |   3339   |      0    |   2000   |      0    |   8678   |      0    | 0         |   2000 |   2000 |  86700    |   2000 |   1 |   1 |
| Linn           |    399   |      0    |      0   |      0    |      0   |      0    |    399   |      0    | 0         |      0 |      0 |      0    |      0 |   0 |   0 |
| Livingston     |   1805   |      0    |   1805   |      0    |   2000   |      0    |   5610   |      0    | 0         |   2000 |   2000 |   2000    |   2000 |   1 |   1 |
| Madison        |  13538   |  15360.8  |  13538   |   2311.79 |  13538   |  10157.6  |  40614   |  27830.2  | 0.685236  |   9477 |   8529 |  96600    |   8529 |   0 |   1 |
| Maries         |  34416   |  11289.4  |  44741   |  36416.9  |  22371   |   7125.51 | 101528   |  54831.8  | 0.540066  |  16778 |  15100 |  80400    |  15100 |   0 |   1 |
| Marion         |      0   |      0    |      0   |      0    |      0   |      0    |      0   |      0    | 0         |      0 |      0 |      0    |      0 |   0 |   0 |
| McDonald       |  18191   |   1725.5  |  18191   |  35954.4  |   9096   |  15683.1  |  45478   |  53363    | 1.17338   |   7732 |   9278 | 176600    |   9278 |   0 |   1 |
| Miller         |  45000   |  13693.5  |  45000   |  29508.1  |  22500   |  10641.4  | 112500   |  53843    | 0.478604  |  16875 |  13500 |  44800    |  13500 |   0 |   1 |
| Mississippi    |      0   |      0    |      0   |      0    |      0   |      0    |      0   |      0    | 0         |      0 |      0 |      0    |      0 |   0 |   0 |
| Moniteau       |  14214   |   4665.17 |  14214   |   8790.48 |   7107   |      0    |  35535   |  13455.6  | 0.378659  |   4975 |   3980 |  54500    |   3980 |   0 |   1 |
| Montgomery     |  13538   |  44579.1  |  13538   |    800    |  15569   |      0    |  42645   |  45379.1  | 1.06411   |  13234 |  15881 |  24000    |  15881 |   0 |   1 |
| Morgan         |  18191   |  14987.6  |  18191   |  14491.9  |   9096   |  30646.2  |  45478   |  60125.7  | 1.32208   |   6822 |   8186 | 442000    |   8186 |   0 |   1 |
| New Madrid     |      0   |      0    |      0   |      0    |      0   |      0    |      0   |      0    | 0         |      0 |      0 |      0    |      0 |   0 |   0 |
| Newton         |  15711   |    400    |  20424   |   3278.46 |  21718.4 |  51294.4  |  57853.4 |  54972.8  | 0.95021   |  15318 |  18382 | 169600    |  18382 |   0 |   1 |
| Nodaway        |  14205   |      0    |  14205   |   2848.34 |   7103   |      0    |  35513   |   2848.34 | 0.0802056 |   4972 |   3978 |   5696.68 |   3978 |   0 |   1 |
| Oregon         |  25000   |  13544.1  |  25000   |      0    |  12500   |  14833.2  |  62500   |  28377.3  | 0.454037  |   8750 |   7000 |  70000    |   7000 |   0 |   1 |
| Osage          |  15750   |   6910.35 |  20475   |  21943    |  23546   |  18745.2  |  59771   |  47598.6  | 0.796349  |  20014 |  18013 |  50400    |  18013 |   0 |   1 |
| Pemiscot       |      0   |      0    |      0   |      0    |      0   |      0    |      0   |      0    | 0         |      0 |      0 |      0    |      0 |   0 |   0 |
| Perry          | 146694   | 130469    | 190702   |  48807.8  | 219307   |  76047.1  | 556703   | 255324    | 0.458635  | 164480 | 131584 | 762000    | 131584 |   0 |   1 |
| Phelps         |  19872.7 |  24937.1  |  23648   |      0    |  27195   |  70369.3  |  70715.7 |  95306.4  | 1.34774   |  19037 |  22844 | 142450    |  22844 |   0 |   1 |
| Pike           |  13538   |      0    |  13538   |      0    |   6769   |      0    |  33845   |      0    | 0         |   4738 |   3790 |  25000    |   3790 |   0 |   1 |
| Platte         |   9025   |      0    |   9025   |      0    |   4513   |      0    |  22563   |      0    | 0         |   2000 |   2000 |   2500    |   2000 |   1 |   1 |
| Polk           |  22173   |  14180.7  |  28825   |   7393.29 |  33149   |  75258.6  |  84147   |  96832.5  | 1.15075   |  24862 |  29834 | 146500    |  29834 |   0 |   1 |
| Pulaski        |  13876   |      0    |  13876   |    800    |   6938   |   7847.02 |  34690   |   8647.02 | 0.249265  |   4857 |   3886 |  45100    |   3886 |   0 |   1 |
| Putnam         |  15000   |   2400    |  19500   |   1200    |   9750   |  18484.1  |  44250   |  22084.1  | 0.499076  |   6825 |   5460 |  22400    |   5460 |   0 |   1 |
| Ralls          |  13538   |      0    |  13538   |      0    |   2000   |      0    |  29076   |      0    | 0         |   2000 |   2000 |  15000    |   2000 |   1 |   1 |
| Randolph       |   3610   |      0    |   3610   |      0    |   1805   |      0    |   9025   |      0    | 0         |   2000 |   2000 |  23600    |   2000 |   1 |   1 |
| Ray            |  13538   |   5137.11 |  13538   |   5361.29 |   6769   |   2000    |  33845   |  12498.4  | 0.369283  |   4738 |   3790 |  11200    |   3790 |   0 |   1 |
| Reynolds       |  13538   |      0    |  35053.4 |  21515.4  |   6769   |   5144.33 |  55360.4 |  26659.7  | 0.481566  |   4738 |   3790 |  84336    |   3790 |   0 |   1 |
| Ripley         |  23009.6 |      0    |  13538   |   7619.72 |   2000   |      0    |  38547.6 |   7619.72 | 0.19767   |   2000 |   2000 |  45000    |   2000 |   1 |   1 |
| Saline         |      0   |      0    |      0   |      0    |      0   |      0    |      0   |      0    | 0         |      0 |   5000 |  25000    |   5000 |   0 |   1 |
| Schuyler       |  15750   |  12800    |  20475   |   2000    |  20475   |    800    |  56700   |  15600    | 0.275132  |  15356 |  12285 |  31700    |  12285 |   0 |   1 |
| Scotland       |  18191   |   6400    |  18191   |   9016.26 |  18191   |   2800    |  54573   |  18216.3  | 0.333796  |  13643 |  10914 |  27000    |  10914 |   0 |   1 |
| Scott          |  13538   |      0    |  13538   |      0    |   2000   |      0    |  29076   |      0    | 0         |   2000 |   2000 |  41695    |   2000 |   1 |   1 |
| Shannon        |  15336   |  17715.6  |  19937   |  14084.4  |  22928   |  48642.5  |  58201   |  80442.6  | 1.38215   |  17196 |  20635 |  80000    |  20635 |   0 |   1 |
| Shelby         |      0   |      0    |      0   |      0    |      0   |      0    |      0   |      0    | 0         |      0 |      0 |      0    |      0 |   0 |   0 |
| St Charles     |   1805   |      0    |   1805   |      0    |    903   |      0    |   4513   |      0    | 0         |   2000 |   2000 |   3800    |   2000 |   1 |   1 |
| St Clair       |  16459   |  19102.4  |  16459   |  23108.8  |   8230   |   8041.16 |  41148   |  50252.3  | 1.22126   |   6173 |   7408 | 116000    |   7408 |   0 |   1 |
| St Francois    |  15337   |  38460.1  |  15337   |  23042.3  |  17638   |      0    |  48312   |  61502.4  | 1.27303   |  14992 |  17990 |  30000    |  17990 |   0 |   1 |
| St Louis       |   5369   |      0    |   5369   |      0    |   2000   |      0    |  12738   |      0    | 0         |   2000 |   2000 |   2300    |   2000 |   1 |   1 |
| Ste Genevieve  |  25407   |  32548.9  |  33029   |  24598.5  |  37983   |  31492.9  |  96419   |  88640.3  | 0.919324  |  28487 |  34184 |  72500    |  34184 |   0 |   1 |
| Stoddard       |   5200   |   2000    |   6760   |    400    |   6760   |    400    |  18720   |   2800    | 0.149573  |   5070 |   4056 |   6760    |   4056 |   0 |   1 |
| Stone          |   7600   |      0    |   7600   |   6612.53 |   3800   |   1027.56 |  19000   |   7640.09 | 0.40211   |   2660 |   2128 |  70500    |   2128 |   1 |   1 |
| Sullivan       |      0   |      0    |  10000   |   1600    |   2000   |      0    |  12000   |   1600    | 0.133333  |      0 |      0 |      0    |      0 |   0 |   0 |
| Taney          |  52250   |   5740.02 |  52250   |  81626.5  |  26125   |   9882.71 | 130625   |  97249.3  | 0.744492  |  19594 |  17635 |  60800    |  17635 |   0 |   1 |
| Texas          |  89306   | 193420    | 116098   | 172639    | 133513   | 234383    | 338917   | 600442    | 1.77165   | 113486 | 136183 | 564600    | 136183 |   0 |   1 |
| Vernon         |  18191   |      0    |  18191   |      0    |   2000   |      0    |  38382   |      0    | 0         |   2000 |   2000 | 122539    |   2000 |   1 |   1 |
| Warren         |  22184   |  19797.9  |  13650   |   5679    |  13650   |      0    |  49484   |  25476.9  | 0.514852  |  10238 |   9214 |  20500    |   9214 |   0 |   1 |
| Washington     |  13538   |      0    |  35180   |  21642    |   2000   |   2739.2  |  50718   |  24381.2  | 0.480721  |   2000 |   2000 |  53111    |   2000 |   1 |   1 |
| Wayne          |  17365   |  87469.8  |  22575   |  42016.1  |  25961   |  38246.2  |  65901   | 167732    | 2.54521   |  22067 |  26480 |  70000    |  26480 |   0 |   1 |
| Webster        |  36100   |  22983.3  |  36100   |  40796    |  36100   |  58761.1  | 108300   | 122540    | 1.13149   |  27075 |  32490 | 157750    |  32490 |   0 |   1 |
| Worth          |      0   |      0    |      0   |      0    |      0   |      0    |      0   |      0    | 0         |      0 |      0 |      0    |      0 |   0 |   0 |
| Wright         |  57486   |  24052.6  |  57486   |  46518.2  |  28743   |  31560.5  | 143715   | 102131    | 0.710652  |  21557 |  19401 | 166700    |  19401 |   0 |   1 |



labeled_formulas: N=(C*O)+L
output: {"Arithmetic-Relationship": []}
source: $MMTU_HOME/resources/Relationship-finding/AR/table_477522___xlsx_633771bf431777a1432a208c720175d0ace9b566.xlsx___line_289