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 |
|----:|:---------------|---------:|--------:|---------:|----------:|-------:|----:|----:|---------:|----:|
|   1 | Accomack       |   360913 |  275475 |   923496 |   1559884 | 504019 |   0 |   0 |    39445 |   1 |
|   2 | Albemarle      |  3604276 |  904088 |  5200619 |   9708983 | 406729 |   0 |   0 |   268425 |   2 |
|   3 | Alleghany      |   676371 |   11300 |   238040 |    925711 |  43433 |   0 |   0 |    34365 |   3 |
|   4 | Amelia         |   278961 |       0 |   348447 |    627408 |  68570 |   0 |   0 |   118392 |   4 |
|   5 | Amherst        |   499083 |       0 |   812489 |   1311572 | 150795 |   0 |   0 |    76252 |   5 |
|   6 | Appomattox     |   199580 |   18750 |   201121 |    419451 |  75439 |   0 |   0 |    21580 |   6 |
|   8 | Augusta        |   943202 |    9500 |  1470245 |   2422947 | 175900 |   0 |   0 |   446913 |   8 |
|   9 | Bath           |   343840 |       0 |   227058 |    570898 |  48375 |   0 |   0 |     9450 |   9 |
|  10 | Bedford        |        0 |       0 |        0 |         0 |      0 |   0 |   0 |        0 |  10 |
|  12 | Botetourt      |  1770266 |   27707 |  1124880 |   2922853 | 164362 |   0 |   0 |   133466 |  12 |
|  13 | Brunswick      |    16163 |  159504 |   272653 |    448320 |  67285 |   0 |   0 |    15479 |  13 |
|  14 | Buchanan       |   774745 |       0 |   684469 |   1459214 | 122576 |   0 |   0 |   155219 |  14 |
|  15 | Buckingham     |   217633 |   15563 |   286898 |    520094 |  51534 |   0 |   0 |    12779 |  15 |
|  16 | Campbell       |   604423 |    5000 |  1115394 |   1724817 | 166578 |   0 |   0 |    89483 |  16 |
|  17 | Caroline       |   436747 |       0 |   514698 |    951445 | 116786 |   0 |   0 |    50269 |  17 |
|  18 | Carroll        |   777085 |       0 |   407720 |   1184805 |  52906 |   0 |   0 |   307910 |  18 |
|  19 | Charles City   |   463339 |   33817 |   133670 |    630826 |   4500 |   0 |   0 |    20503 |  19 |
|  20 | Charlotte      |    10052 |   79448 |   271461 |    360961 |  53754 |   0 |   0 |        0 |  20 |
|  21 | Chesterfield   |  9925757 |  844988 |  9744180 |  20514925 | 219066 |   0 |   0 |  1011969 |  21 |
|  22 | Clarke         |   790945 |   20750 |   241150 |   1052845 |  12730 |   0 |   0 |   324565 |  22 |
|  23 | Craig          |        0 |    1200 |    38374 |     39574 |   8332 |   0 |   0 |        0 |  23 |
|  24 | Culpeper       |  1218240 |       0 |  1144433 |   2362673 | 161944 |   0 |   0 |   160308 |  24 |
|  25 | Cumberland     |    53262 |       0 |   115450 |    168712 |      0 |   0 |   0 |    19102 |  25 |
|  26 | Dickenson      |   155198 |       0 |   435569 |    590767 |  89768 |   0 |   0 |    27682 |  26 |
|  27 | Dinwiddie      |   862073 |       0 |   287395 |   1149468 |      0 |   0 |   0 |   115970 |  27 |
|  28 | Essex          |   263009 |   23000 |   200059 |    486068 |      0 |   0 |   0 |    25908 |  28 |
|  29 | Fairfax        | 86603942 | 5202188 | 40020855 | 131826985 | 526606 |   0 |   0 | 47732711 |  29 |
|  30 | Fauquier       |  4159572 |       0 |  2583871 |   6743443 | 171465 |   0 |   0 |   381828 |  30 |
|  31 | Floyd          |   123141 |       0 |   255054 |    378195 |      0 |   0 |   0 |    31467 |  31 |
|  33 | Franklin       |  1471298 |       0 |   965876 |   2437174 | 164929 |   0 |   0 |   177919 |  33 |
|  34 | Frederick      |  6407353 |       0 |  1218200 |   7625553 | 190846 |   0 |   0 |  1923665 |  34 |
|  35 | Giles          |   531282 |       0 |    29250 |    560532 |      0 |   0 |   0 |   339311 |  35 |
|  37 | Goochland      |   653812 |       0 |   512689 |   1166501 |  56854 |   0 |   0 |    78233 |  37 |
|  39 | Greene         |   183117 |       0 |   463557 |    646674 |  35215 |   0 |   0 |    44186 |  39 |
|  40 | Greensville    |   203263 |   10000 |   177967 |    391230 |  44292 |   0 |   0 |     4042 |  40 |
|  41 | Halifax        |   225297 |   15000 |   330806 |    571103 |  90797 |   0 |   0 |    18811 |  41 |
|  42 | Hanover        |  3558882 |       0 |  3202464 |   6761346 | 349563 |   0 |   0 |   525628 |  42 |
|  44 | Henry          |  1339712 |  103588 |   735541 |   2178841 |      0 |   0 |   0 |   177542 |  44 |
|  45 | Highland       |    46864 |       0 |    70440 |    117304 |      0 |   0 |   0 |    45981 |  45 |
|  48 | King & Queen   |        0 |    3712 |   194619 |    198331 |  21687 |   0 |   0 |     1081 |  48 |
|  50 | King William   |   366924 |    9500 |   496125 |    872549 |  59618 |   0 |   0 |   175819 |  50 |
|  51 | Lancaster      |    50004 |    5043 |   133000 |    188047 |      0 |   0 |   0 |        0 |  51 |
|  52 | Lee            |        0 |       0 |        0 |         0 |      0 |   0 |   0 |        0 |  52 |
|  54 | Louisa         |  1129696 |   75400 |   432378 |   1637474 |  37683 |   0 |   0 |   467707 |  54 |
|  55 | Lunenburg      |        0 |       0 |        0 |         0 |      0 |   0 |   0 |        0 |  55 |
|  56 | Madison        |   535000 |    3000 |   143600 |    681600 |      0 |   0 |   0 |   106199 |  56 |
|  57 | Mathews        |    96417 |       0 |   358181 |    454598 |  96929 |   0 |   0 |     3640 |  57 |
|  58 | Mecklenburg    |    12600 |   70000 |   653522 |    736122 | 142327 |   0 |   0 |    26843 |  58 |
|  59 | Middlesex      |   165075 |       0 |   115000 |    280075 |   4951 |   0 |   0 |    38733 |  59 |
|  60 | Montgomery     |  1046911 |  109320 |  2764679 |   3920910 | 532125 |   0 |   0 |   497903 |  60 |
|  61 | Nelson         |   196267 |    9000 |   334239 |    539506 |  30640 |   0 |   0 |    44878 |  61 |
|  62 | New Kent       |   410603 |   23900 |   179956 |    614459 |   4500 |   0 |   0 |   249125 |  62 |
|  63 | Northampton    |   241389 |       0 |   349212 |    590601 | 236188 |   0 |   0 |    52280 |  63 |
|  64 | Northumberland |    50000 |       0 |   157600 |    207600 |   1436 |   0 |   0 |        0 |  64 |
|  65 | Nottoway       |    21500 |       0 |   249609 |    271109 |  57213 |   0 |   0 |      225 |  65 |
|  66 | Orange         |   228210 |   43000 |  1052344 |   1323554 | 163363 |   0 |   0 |    78311 |  66 |
|  67 | Page           |   108831 |       0 |   385603 |    494434 |  52276 |   0 |   0 |    54387 |  67 |
|  68 | Patrick        |   221740 |       0 |   280782 |    502522 |   4500 |   0 |   0 |    14465 |  68 |
|  69 | Pittsylvania   |   448170 |       0 |  1605295 |   2053465 | 167646 |   0 |   0 |    68191 |  69 |
|  70 | Powhatan       |  1517324 |       0 |   556172 |   2073496 | 126581 |   0 |   0 |    33914 |  70 |
|  71 | Prince Edward  |    58000 |   13285 |   360528 |    431813 |  59105 |   0 |   0 |        0 |  71 |
|  72 | Prince George  |   970447 |    8882 |   604127 |   1583456 |      0 |   0 |   0 |    68495 |  72 |
|  73 | Prince William | 37296000 |  896000 | 15835000 |  54027000 | 525000 |   0 |   0 |  9470000 |  73 |
|  74 | Pulaski        |   637638 |   34860 |   753536 |   1426034 | 137843 |   0 |   0 |   190838 |  74 |
|  75 | Rappahannock   |    56531 |       0 |   286036 |    342567 |  48886 |   0 |   0 |        0 |  75 |
|  76 | Richmond       |    43500 |   -3626 |   100160 |    140034 |      0 |   0 |   0 |        0 |  76 |
|  78 | Rockbridge     |   960461 |       0 |   912341 |   1872802 | 193291 |   0 |   0 |   112522 |  78 |
|  79 | Rockingham     |  1639406 |       0 |  1455469 |   3094875 | 220156 |   0 |   0 |   666314 |  79 |
|  80 | Russell        |   260914 |       0 |   348625 |    609539 |  88010 |   0 |   0 |     5880 |  80 |
|  81 | Scott          |   206498 |   87056 |   278547 |    572101 |  62312 |   0 |   0 |    63637 |  81 |
|  82 | Shenandoah     |  1578372 |   16250 |   792159 |   2386781 |   4500 |   0 |   0 |   910056 |  82 |
|  83 | Smyth          |    28991 |       0 |   865792 |    894783 | 158827 |   0 |   0 |        0 |  83 |
|  84 | Southampton    |    15000 |   38485 |   361668 |    415153 |  79185 |   0 |   0 |     6218 |  84 |
|  85 | Spotsylvania   |  3043092 |   86727 |  5199453 |   8329272 | 263141 |   0 |   0 |   277058 |  85 |
|  86 | Stafford       |  6992997 |  348858 |  6355938 |  13697793 | 319627 |   0 |   0 |   826838 |  86 |
|  87 | Surry          |   478457 |       0 |   150398 |    628855 |   4996 |   0 |   0 |    16943 |  87 |
|  88 | Sussex         |    45000 |   15000 |   233810 |    293810 |  48283 |   0 |   0 |     4020 |  88 |
|  89 | Tazewell       |   201807 |   76479 |   983476 |   1261762 | 165890 |   0 |   0 |    15530 |  89 |
|  90 | Warren         |  2782250 |   90853 |   991883 |   3864986 |  12711 |   0 |   0 |   289824 |  90 |
|  93 | Wise           |    45000 |   66105 |  1045961 |   1157066 | 224239 |   0 |   0 |    33211 |  93 |



label: F=(C+D+E)
labeled_formulas: I=A*H
K=A+H
I=C*H
I=D*H
I=E*H
I=F*H
I=G*H
I=H*J
I=H*K
F=(C+D+E)
I=(A+C)*H
H=(A-K)*C
I=(A*D)*H
K=(D*H)+A
I=(A+E)*H
H=(A-K)*E
I=(A+F)*H
H=(A-K)*F
I=(A+G)*H
H=(A-K)*G
I=(A*H)*J
K=(H*J)+A
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=(D*E)*H
I=(D*F)*H
I=(D*G)*H
I=(D*H)*J
I=(D*H)*K
I=(E+F)*H
I=(E+G)*H
I=(E*H)*J
I=(E*H)*K
I=(F+G)*H
I=(F*H)*J
I=(F*H)*K
I=(G*H)*J
I=(G*H)*K
I=(H*J)*K
output: {"Arithmetic-Relationship": ["F=(C+D+E)"]}
source: $MMTU_HOME/resources/Relationship-finding/AR/table_2325___xlsx_756f2de2f1b7907d0304e13012e31a03c6e2b186.xlsx___line_334