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 |
|:------------------------|----------:|----------:|-----------:|----:|-----------:|----:|----:|---------:|-----------:|-----:|
| ACCOMACK                |  3130.49  |  16247.2  |   9837.62  |   0 |    841.147 |   0 |   0 |  133.89  |   5434.51  |  680 |
| ALBEMARLE               |  7874.92  |  28707.9  |  15850.9   |   0 |   4086.01  |   0 |   0 |  814.539 |   7956.41  | 1098 |
| ALLEGHANY               |  1484.78  |   7383.56 |   2196.74  |   0 |    137.307 |   0 |   0 |   23.051 |   5026.47  |  526 |
| AMELIA                  |  1137.45  |  11517.9  |   4357.06  |   0 |    262.501 |   0 |   0 |  188.837 |   6709.53  |  366 |
| AMHERST                 |  2543.49  |  15759.7  |   8021.88  |   0 |    802.176 |   0 |   0 |   95.343 |   6840.26  |  728 |
| APPOMATTOX              |  1190.8   |   8055.1  |   4038.93  |   0 |    496.936 |   0 |   0 |   48.511 |   3470.72  |  369 |
| ARLINGTON               | 13540.8   |  80107.9  |  44149.5   |   0 |  23050.1   |   0 |   0 | 1940.01  |  10968.4   | 1444 |
| AUGUSTA                 |  5528.32  |  25815.5  |  11529.8   |   0 |    863.892 |   0 |   0 |  233.434 |  13188.4   | 1650 |
| BATH                    |   427.712 |   2644.13 |   1556.94  |   0 |    159.706 |   0 |   0 |   10.846 |    916.638 |  114 |
| BEDFORD                 |  6464.91  |  46447.8  |  19806.1   |   0 |   2005.83  |   0 |   0 |  278.583 |  24357.3   | 2091 |
| BLAND                   |   496.196 |   5083.7  |   2876.85  |   0 |     82.009 |   0 |   0 |   20.815 |   2104.03  |  233 |
| BOTETOURT               |  2841.35  |  23688.5  |   9269.47  |   0 |    572.168 |   0 |   0 |   63.951 |  13782.9   | 1029 |
| BRUNSWICK               |  1124.51  |  11794.6  |   5574.6   |   0 |    280.29  |   0 |   0 |   74.38  |   5865.35  |  409 |
| BUCHANAN                |   917.809 |  12030.5  |   8707.33  |   0 |    257.764 |   0 |   0 |  102.948 |   2962.45  |  383 |
| BUCKINGHAM              |  1124.26  |   7077.13 |   3464.8   |   0 |    145.8   |   0 |   0 |   68.569 |   3397.97  |  280 |
| CAMPBELL                |  4684.26  |  22566.2  |   9272.54  |   0 |    810.083 |   0 |   0 |   77.97  |  12405.6   | 1318 |
| CAROLINE                |  2776.61  |  24194.3  |  11857.8   |   0 |   1666.1   |   0 |   0 |  103.067 |  10567.4   |  751 |
| CARROLL                 |  2265.44  |  17021.8  |   6394.19  |   0 |    281.225 |   0 |   0 |   64.227 |  10282.2   |  956 |
| CHARLES CITY            |   649.503 |   5962.34 |   1871.37  |   0 |     84.862 |   0 |   0 |  217.476 |   3788.63  |  225 |
| CHARLOTTE               |   868.848 |   7870.05 |   4423.67  |   0 |    203.482 |   0 |   0 |   50.057 |   3192.84  |  336 |
| CHESTERFIELD            | 27471.1   | 245454    | 123110     |   0 |  20415.6   |   0 |   0 | 1906.47  | 100022     | 7523 |
| CLARKE                  |  1340.82  |   7585.69 |   3429.16  |   0 |    669.707 |   0 |   0 |  177.119 |   3309.71  |  282 |
| CRAIG                   |   519.952 |   4889.63 |   2013.66  |   0 |    142.426 |   0 |   0 |    0.616 |   2732.93  |  184 |
| CUMBERLAND              |   731.209 |   6300.39 |   1565.85  |   0 |     84.937 |   0 |   0 |    0.227 |   4649.38  |  284 |
| DICKENSON               |   904.602 |  11464.2  |   6116.7   |   0 |    117.346 |   0 |   0 |    2.365 |   5227.77  |  459 |
| DINWIDDIE               |  2619.28  |  18408.6  |   7137.15  |   0 |    693.007 |   0 |   0 |   48.185 |  10530.2   |  861 |
| ESSEX                   |   857.306 |   7012.37 |   3391.38  |   0 |    182.217 |   0 |   0 |   43.579 |   3395.19  |  253 |
| FAIRFAX                 | 74073.4   | 469859    | 293849     |   0 | 122460     |   0 |   0 | 8159.27  |  45391.2   | 7442 |
| FAUQUIER                |  6136.76  |  24519.2  |  15814     |   0 |   3526.92  |   0 |   0 |  422.211 |   4756.05  |  677 |
| FLOYD                   |  1298.82  |   7472.9  |   3095.8   |   0 |    168.861 |   0 |   0 |    2.026 |   4206.22  |  390 |
| FLUVANNA                |  2390.64  |  10881.9  |   6215.15  |   0 |    819.097 |   0 |   0 |  213.137 |   3634.47  |  484 |
| FRANKLIN                |  4407.66  |  37951.4  |  17160.8   |   0 |    778.515 |   0 |   0 |  245.424 |  19766.7   | 1630 |
| FREDERICK               |  7001.27  |  42243    |  20096.7   |   0 |   3425.47  |   0 |   0 |  465.105 |  18255.8   | 1818 |
| GILES                   |  1466.65  |  10582.3  |   5881.7   |   0 |    231.605 |   0 |   0 |   44.189 |   4424.8   |  381 |
| GLOUCESTER              |  4588.93  |  33688    |  21487.1   |   0 |   3918.58  |   0 |   0 |  123.839 |   8158.47  |  761 |
| GOOCHLAND               |  1715.7   |   9379.21 |   3876.38  |   0 |    359.503 |   0 |   0 |   67.099 |   5076.23  |  371 |
| GRAYSON                 |  1096.01  |   7680.66 |   3706.37  |   0 |    109.377 |   0 |   0 |   29.664 |   3835.25  |  521 |
| GREENE                  |  1871.03  |   9253.03 |   5321.1   |   0 |   1085.38  |   0 |   0 |   75.234 |   2771.31  |  313 |
| GREENSVILLE             |   737.852 |   5190.25 |   1579.87  |   0 |     57.182 |   0 |   0 |   21.731 |   3531.46  |  259 |
| HALIFAX                 |  2619.53  |  25750.5  |  13314.1   |   0 |    559.654 |   0 |   0 |   57.946 |  11818.8   |  981 |
| HANOVER                 |  7570.91  |  42835.7  |  20604     |   0 |   3132.24  |   0 |   0 |  658.527 |  18440.9   | 1410 |
| HENRY                   |  4033.28  |  29281.5  |  10519.9   |   0 |    719.348 |   0 |   0 |   96.799 |  17945.4   | 1749 |
| HIGHLAND                |   211.674 |   1268.36 |    673.879 |   0 |     24.671 |   0 |   0 |   20.96  |    548.845 |   64 |
| ISLE OF WIGHT           |  4184.77  |  31659.5  |  19865.3   |   0 |   5280.45  |   0 |   0 |  122.123 |   6391.58  |  726 |
| JAMES CITY              |  9068.29  |  75821.9  |  51635.1   |   0 |  12101.4   |   0 |   0 | 1156.7   |  10928.8   | 1267 |
| KING AND QUEEN          |   566.793 |   4349.02 |   1856.35  |   0 |    157.057 |   0 |   0 |   34.925 |   2300.69  |  155 |
| KING GEORGE             |  3140.34  |  24649    |  14374.7   |   0 |   3527.18  |   0 |   0 |   82.467 |   6664.66  |  577 |
| KING WILLIAM            |  1296.23  |   7736.41 |   4223.9   |   0 |    614.632 |   0 |   0 |   19     |   2878.88  |  247 |
| LANCASTER               |  1204.81  |   7921.12 |   4427.16  |   0 |    124.775 |   0 |   0 |  158.595 |   3210.59  |  264 |
| LEE                     |  1733.38  |  16931.2  |   9488.97  |   0 |    181.322 |   0 |   0 |   40.518 |   7220.37  |  688 |
| LOUDOUN                 | 20283     | 106203    |  64517.1   |   0 |  26672     |   0 |   0 | 1477.48  |  13536.1   | 2062 |
| LOUISA                  |  3307.27  |  17010.7  |   8719.57  |   0 |    709.797 |   0 |   0 |  207.725 |   7373.6   |  662 |
| LUNENBURG               |  1087.93  |   9902.57 |   3563.43  |   0 |    155.339 |   0 |   0 |  111.726 |   6072.07  |  370 |
| MADISON                 |  1027.73  |   5153.66 |   2831.76  |   0 |    318.098 |   0 |   0 |  199.762 |   1804.05  |  228 |
| MATHEWS                 |  1035.64  |   5540.52 |   3711.95  |   0 |    327.422 |   0 |   0 |  105.876 |   1395.27  |  165 |
| MECKLENBURG             |  2789.42  |  25664.7  |  11995.1   |   0 |    682.151 |   0 |   0 |  253.838 |  12733.5   |  844 |
| MIDDLESEX               |  1154.8   |   7420.91 |   4492.08  |   0 |    369.835 |   0 |   0 |   76.861 |   2482.13  |  259 |
| MONTGOMERY              |  4508.53  |  31111.3  |  14616     |   0 |   3642.97  |   0 |   0 |  598.711 |  12253.7   | 1173 |
| NELSON                  |  1103.75  |   7753.35 |   4707.82  |   0 |    421.768 |   0 |   0 |  135.279 |   2488.48  |  324 |
| NEW KENT                |  1900.5   |  18110.8  |  10218.1   |   0 |   1078.14  |   0 |   0 |  334.633 |   6479.96  |  504 |
| NORTHAMPTON             |  1154.48  |   5434.84 |   3312.32  |   0 |    405.021 |   0 |   0 |   33.628 |   1683.88  |  205 |
| NORTHUMBERLAND          |  1431.15  |   6621.21 |   2868.68  |   0 |    226.034 |   0 |   0 |  203.555 |   3322.94  |  305 |
| NOTTOWAY                |  1139.45  |  12528.8  |   6044.22  |   0 |    465.069 |   0 |   0 |   28.708 |   5990.82  |  445 |
| ORANGE                  |  3444.59  |  22535.4  |  14381     |   0 |   2847.62  |   0 |   0 |  251.328 |   5055.47  |  656 |
| PAGE                    |  1890.77  |  11201.9  |   5732.28  |   0 |    294.707 |   0 |   0 |  287.831 |   4887.05  |  567 |
| PATRICK                 |  1357.85  |  11023.3  |   5290.17  |   0 |    154.842 |   0 |   0 |    7.612 |   5570.7   |  588 |
| PITTSYLVANIA            |  4916.78  |  32209.8  |  13534.8   |   0 |   1018.06  |   0 |   0 |  102.112 |  17554.9   | 2014 |
| POWHATAN                |  2048.99  |  13856.1  |   6803.33  |   0 |    911.816 |   0 |   0 |   89.849 |   6051.11  |  468 |
| PRINCE EDWARD           |  1572.51  |  13355.4  |   6285.97  |   0 |    642.718 |   0 |   0 |  211.235 |   6215.46  |  485 |
| PRINCE GEORGE           |  5076.25  |  48023.3  |  24027.8   |   0 |   6456.4   |   0 |   0 |  205.466 |  17333.7   | 1558 |
| PULASKI                 |  2800.5   |  17958.4  |   8678.5   |   0 |    408.797 |   0 |   0 |  231.766 |   8639.37  |  913 |
| RAPPAHANNOCK            |   664.095 |   2770.71 |   1085.43  |   0 |     89.991 |   0 |   0 |   69.355 |   1525.93  |  109 |
| RICHMOND                |   546.495 |   4836.12 |   2459.43  |   0 |    177.42  |   0 |   0 |   22.719 |   2176.55  |  151 |
| ROANOKE                 |  8076.01  |  70894.2  |  27294.1   |   0 |   2283.46  |   0 |   0 | 1043.04  |  40273.6   | 2736 |
| ROCKBRIDGE              |  1952.52  |  11468.7  |   3676.83  |   0 |    281.012 |   0 |   0 |   16.094 |   7494.78  |  602 |
| ROCKINGHAM              |  4540.03  |  21980.8  |  11054.3   |   0 |    802.031 |   0 |   0 |  468.299 |   9656.15  | 1520 |
| RUSSELL                 |  1360.48  |  11182.5  |   7010.76  |   0 |    260.06  |   0 |   0 |   58.547 |   3853.16  |  486 |
| SCOTT                   |  1476.4   |  13454    |   8061.18  |   0 |    293.711 |   0 |   0 |   35.162 |   5063.94  |  608 |
| SHENANDOAH              |  3625.01  |  24926    |  11496.1   |   0 |   1172.23  |   0 |   0 |  263.707 |  11994     | 1172 |
| SMYTH                   |  2298.96  |  18560.7  |  11150.1   |   0 |    381.628 |   0 |   0 |   68.73  |   6960.31  |  821 |
| SOUTHAMPTON             |  1567.81  |   9423.99 |   4613.21  |   0 |    634.189 |   0 |   0 |    8.72  |   4167.87  |  331 |
| SPOTSYLVANIA            | 14496     | 114466    |  65827.8   |   0 |  18716.5   |   0 |   0 |  693.618 |  29228.3   | 3309 |
| STAFFORD                | 21484     | 190613    | 123506     |   0 |  42968.6   |   0 |   0 |  800.013 |  23337.9   | 3865 |
| SURRY                   |   666.264 |   4785.49 |   2479.02  |   0 |    315.09  |   0 |   0 |   51.974 |   1939.41  |  181 |
| SUSSEX                  |   827.593 |   7481.79 |   3818.72  |   0 |    380.839 |   0 |   0 |   10     |   3272.23  |  289 |
| TAZEWELL                |  2842.13  |  35576    |  21874.9   |   0 |    710.819 |   0 |   0 |  300.809 |  12689.5   | 1295 |
| WARREN                  |  3497.79  |  20647.4  |   9735.02  |   0 |   1564.89  |   0 |   0 |   92.244 |   9255.22  |  886 |
| WASHINGTON              |  4226.05  |  33509.4  |  18504.4   |   0 |    827.402 |   0 |   0 |  304.722 |  13872.8   | 1565 |
| WESTMORELAND            |  1865.66  |  10872.1  |   5298.76  |   0 |    767.912 |   0 |   0 |   31.511 |   4773.91  |  428 |
| WISE                    |  2581.56  |  24839.8  |  16101.7   |   0 |    896.259 |   0 |   0 |  283.307 |   7558.52  |  908 |
| WYTHE                   |  2258.21  |  18224.4  |   9455.62  |   0 |    399.212 |   0 |   0 |  137.099 |   8232.49  |  862 |
| YORK                    |  9698.37  |  70263.5  |  43458.7   |   0 |  16340.2   |   0 |   0 |  455.655 |  10008.9   | 1389 |
| BRISTOL (CITY)          |  1368.96  |  11959.4  |   5841.28  |   0 |    186.662 |   0 |   0 |   74.777 |   5856.7   |  563 |
| BUENA VISTA (CITY)      |   401.86  |   3840.94 |   1801.44  |   0 |    150.761 |   0 |   0 |    1.685 |   1887.06  |  159 |
| CHARLOTTESVILLE (CITY)  |  1867.83  |  12856    |   5471.33  |   0 |   4962.98  |   0 |   0 |  574.843 |   1846.81  |  318 |
| CHESAPEAKE (CITY)       | 31178.6   | 269751    | 168182     |   0 |  59616.9   |   0 |   0 | 1417.13  |  40535.2   | 5310 |
| COLONIAL HEIGHTS (CITY) |  1897.51  |  28549.7  |  19234     |   0 |   2607.79  |   0 |   0 |  170.78  |   6537.1   |  544 |
| COVINGTON (CITY)        |   455.318 |   7615.16 |   5198.81  |   0 |    146.108 |   0 |   0 |   63.843 |   2206.4   |  200 |
| EMPORIA (CITY)          |   366.032 |   5829.72 |   3564.76  |   0 |    297.252 |   0 |   0 |   19.769 |   1947.93  |  149 |
| FAIRFAX (CITY)          |  1708.43  |  17195.4  |  10199     |   0 |   5836.79  |   0 |   0 |  189.006 |    970.603 |  133 |
| FALLS CHURCH (CITY)     |   746.783 |   5022.47 |   3027.85  |   0 |   1320.84  |   0 |   0 |  120.955 |    552.82  |   65 |
| FRANKLIN (CITY)         |   741.704 |   6860.54 |   4194.61  |   0 |    710.033 |   0 |   0 |   72.951 |   1882.94  |  173 |
| FREDERICKSBURG (CITY)   |  2144.25  |  17594.7  |   9563.2   |   0 |   3160.43  |   0 |   0 |  292.578 |   4578.45  |  522 |
| GALAX (CITY)            |   433.782 |   6253.24 |   4459.35  |   0 |    173.948 |   0 |   0 |   11.306 |   1608.63  |  151 |
| HARRISONBURG (CITY)     |  1521.27  |  11724.5  |   6674.76  |   0 |   1664.26  |   0 |   0 |  279.34  |   3106.15  |  416 |
| HOPEWELL (CITY)         |  2215.67  |  35265.8  |  20742.1   |   0 |   2860.1   |   0 |   0 |  110.881 |  11552.7   |  829 |
| LEXINGTON (CITY)        |   244.82  |   5761.75 |   4366.08  |   0 |    411.221 |   0 |   0 |  219.259 |    765.186 |   87 |
| LYNCHBURG (CITY)        |  4676.09  |  42389.8  |  20106.2   |   0 |   9054.39  |   0 |   0 |  394.496 |  12834.7   | 1519 |
| MANASSAS (CITY)         |  2563.88  |  15574.1  |   9614.79  |   0 |   2972.23  |   0 |   0 |  137.894 |   2849.22  |  283 |
| MANASSAS PARK (CITY)    |   920.049 |   1449.07 |     74.3   |   0 |      0     |   0 |   0 |    0     |   1374.77  |  118 |
| MARTINSVILLE (CITY)     |  1060.13  |  13377.4  |   7764.32  |   0 |    592.061 |   0 |   0 |  110.43  |   4910.61  |  427 |
| NEWPORT NEWS (CITY)     | 24684.9   | 221166    | 123129     |   0 |  42501.7   |   0 |   0 |  897.292 |  54637.6   | 5425 |
| NORFOLK (CITY)          | 30327.7   | 230076    | 118982     |   0 |  59372.5   |   0 |   0 | 1489.8   |  50231.2   | 5644 |
| NORTON (CITY)           |   235.993 |   2369.18 |   1817.67  |   0 |     79.163 |   0 |   0 |   10.796 |    461.552 |   63 |
| PETERSBURG (CITY)       |  3689.67  |  65331.8  |  38824     |   0 |   4535.91  |   0 |   0 |  239.814 |  21732     | 1385 |
| POQUOSON (CITY)         |  1472.14  |  10669.4  |   7476.85  |   0 |   1888.94  |   0 |   0 |   64.508 |   1239.11  |  160 |
| PORTSMOUTH (CITY)       | 11751.5   | 101590    |  55704.4   |   0 |  16413.9   |   0 |   0 |  379.654 |  29091.7   | 2736 |
| RADFORD (CITY)          |   804.122 |   6881.86 |   4185.79  |   0 |    824.367 |   0 |   0 |   51.478 |   1820.23  |  192 |
| STAUNTON (CITY)         |  1813.22  |  17691.7  |   9553.11  |   0 |   1067.1   |   0 |   0 |  429.882 |   6641.65  |  596 |
| SUFFOLK (CITY)          | 11359.3   | 105260    |  65493.1   |   0 |  20996.3   |   0 |   0 |  928.941 |  17841.2   | 2227 |
| VIRGINIA BEACH (CITY)   | 64660.5   | 517112    | 320637     |   0 | 131961     |   0 |   0 | 3895.44  |  60618.4   | 9483 |
| WAYNESBORO (CITY)       |  1669.34  |  12592.6  |   7791.84  |   0 |    713.969 |   0 |   0 |  503.281 |   3583.54  |  419 |
| WILLIAMSBURG (CITY)     |  1042.69  |   2317.69 |    631.14  |   0 |    162.829 |   0 |   0 |    4.506 |   1519.22  |  162 |
| WINCHESTER (CITY)       |  1939.19  |  11296.9  |   4653.09  |   0 |   1154.05  |   0 |   0 |  204.451 |   5285.29  |  546 |



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_393811___xlsx_95c16b37c9755f368b35fdb8f0a57b1b6f1f815e.xlsx___line_561