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 |   Q |
|----:|:---------------|---------:|---------:|---------:|--------:|---------:|---------:|---------:|----:|----:|---------:|---------:|----:|---------:|----------:|----:|
|   1 | Accomack       |  4398808 |  1109000 |    55326 |   62950 |   586500 |     9849 |   415123 |   0 |   0 |   625177 |        0 |   0 |    20872 |   7283605 |   1 |
|   3 | Alleghany      |   983230 |   430803 |   456121 |       0 |   409683 |        0 |    84970 |   0 |   0 |   157418 |   456006 |   0 |     5699 |   2983930 |   3 |
|   4 | Amelia         |   961994 |   240836 |   324093 |       0 |   384918 |    62668 |   155266 |   0 |   0 |        0 |        0 |   0 |    41972 |   2171747 |   4 |
|   5 | Amherst        |  3227313 |   824627 |   383748 |       0 |   793662 |    66237 |   270883 |   0 |   0 |    54573 |  1028064 |   0 |        0 |   6649107 |   5 |
|   6 | Appomattox     |  1455129 |   389629 |        0 |   13325 |   525051 |        0 |   127953 |   0 |   0 |     9019 |        0 |   0 |    45698 |   2565804 |   6 |
|   8 | Augusta        |  6822265 |  1812435 |  4313711 |  267038 |        0 |   355258 |   950812 |   0 |   0 |   451493 |  2418593 |   0 |    70316 |  17461921 |   8 |
|   9 | Bath           |   704233 |    18700 |        0 |       0 |    74597 |    42637 |    40177 |   0 |   0 |  1709035 |   576030 |   0 |     1655 |   3167064 |   9 |
|  10 | Bedford        |        0 |        0 |        0 |       0 |        0 |        0 |        0 |   0 |   0 |        0 |        0 |   0 |        0 |         0 |  10 |
|  11 | Bland          |   471061 |   163015 |        0 |    1686 |   118878 |        0 |    22910 |   0 |   0 |    15575 |   181174 |   0 |    25986 |   1000285 |  11 |
|  12 | Botetourt      |  3281555 |   584390 |   976576 |   57344 |   657354 |   199764 |   416941 |   0 |   0 |   411396 |  1380403 |   0 |   151486 |   8117209 |  12 |
|  13 | Brunswick      |   941604 |   298771 |        0 |   14957 |   362633 |    17510 |    99656 |   0 |   0 |     3681 |        0 |   0 |        0 |   1738812 |  13 |
|  15 | Buckingham     |   863034 |   361905 |        0 |   47375 |   349277 |    56671 |    95398 |   0 |   0 |     3499 |        0 |   0 |        0 |   1777159 |  15 |
|  16 | Campbell       |  5585583 |  1071311 |  2218283 |       0 |  1652940 |   307826 |   584593 |   0 |   0 |   149784 |  1479268 |   0 |        0 |  13049588 |  16 |
|  17 | Caroline       |  2855615 |   633744 |   890641 |       0 |  1050059 |    89619 |   379097 |   0 |   0 |   140435 |  1192417 |   0 |        0 |   7231627 |  17 |
|  18 | Carroll        |  2120199 |   719227 |        0 |       0 |   743739 |    18429 |   177752 |   0 |   0 |   282798 |   526538 |   0 |    97099 |   4685781 |  18 |
|  19 | Charles City   |   949165 |   156212 |        0 |    1200 |       30 |        0 |    82443 |   0 |   0 |     4026 |        0 |   0 |    36228 |   1229304 |  19 |
|  20 | Charlotte      |   753317 |   233157 |        0 |       0 |   325653 |        0 |    64679 |   0 |   0 |        0 |        0 |   0 |    32437 |   1409243 |  20 |
|  21 | Chesterfield   | 54993090 |  8283882 | 24487081 |       0 | 15072427 |  2388577 |  6249709 |   0 |   0 |  4528009 |        0 |   0 |   649624 | 116652399 |  21 |
|  22 | Clarke         |  1281009 |   386597 |    20940 |       0 |   296285 |        0 |   331018 |   0 |   0 |    29702 |        0 |   0 |        0 |   2345551 |  22 |
|  23 | Craig          |   219544 |   118093 |        0 |    4029 |        0 |        0 |    28666 |   0 |   0 |    16142 |    76436 |   0 |    12924 |    475834 |  23 |
|  24 | Culpeper       |  7144442 |   835781 |        0 |       0 |   798671 |        0 |   962595 |   0 |   0 |    51875 |        0 |   0 |        0 |   9793364 |  24 |
|  25 | Cumberland     |   608467 |   177328 |   110847 |       0 |   241117 |     7727 |   122965 |   0 |   0 |        0 |        0 |   0 |        0 |   1268451 |  25 |
|  28 | Essex          |  1962334 |   233025 |        0 |       0 |   403691 |        0 |    27727 |   0 |   0 |        0 |        0 |   0 |        0 |   2626777 |  28 |
|  30 | Fauquier       | 10533415 |  1748470 |  1801247 |       0 |  2011001 |   160441 |  2271130 |   0 |   0 |    62989 |        0 |   0 |        0 |  18588693 |  30 |
|  31 | Floyd          |  1107536 |   369071 |        0 |   45073 |   407957 |        0 |     4168 |   0 |   0 |    59470 |   179851 |   0 |    55370 |   2228496 |  31 |
|  32 | Fluvanna       |  2207343 |   554834 |        0 |       0 |   885451 |    84931 |   435156 |   0 |   0 |        0 |        0 |   0 |        0 |   4167715 |  32 |
|  33 | Franklin       |  5664739 |   999577 |     3633 |  209174 |  2030181 |   156476 |   691985 |   0 |   0 |   144516 |  1176654 |   0 |        0 |  11076935 |  33 |
|  34 | Frederick      | 16037305 |  3773932 |  8589974 |       0 |  2665730 |   532223 |  2068695 |   0 |   0 |   782838 |  5044155 |   0 |    31498 |  39526350 |  34 |
|  35 | Giles          |  1552683 |   244459 |        0 |    5521 |   199639 |        0 |    86617 |   0 |   0 |   105718 |        0 |   0 |    46494 |   2241131 |  35 |
|  36 | Gloucester     |  5469912 |   870900 |  1867032 |       0 |        0 |   303677 |   603745 |   0 |   0 |   184993 |  2346997 |   0 |        0 |  11647256 |  36 |
|  37 | Goochland      |  3883969 |   513506 |  1030634 |       0 |    17560 |   761956 |   946002 |   0 |   0 |     9940 |        0 |   0 |    87843 |   7251410 |  37 |
|  38 | Grayson        |   564237 |   344259 |        0 |       0 |   347793 |    15114 |   123166 |   0 |   0 |    26506 |        0 |   0 |    36338 |   1457413 |  38 |
|  39 | Greene         |  2483454 |   435985 |   664658 |       0 |   473311 |    47509 |   338729 |   0 |   0 |   350413 |   736245 |   0 |        0 |   5530304 |  39 |
|  40 | Greensville    |   725359 |   297698 |   473598 |   37376 |   205051 |        0 |    35979 |   0 |   0 |    73747 |   260308 |   0 |    43111 |   2152227 |  40 |
|  41 | Halifax        |  3470618 |  1027099 |   347415 |   39182 |  1087186 |        0 |   222979 |   0 |   0 |   188282 |   261220 |   0 |   910582 |   7554563 |  41 |
|  42 | Hanover        | 24952021 |  2069167 |  1160246 |  507474 |        0 |  1030391 |  2950008 |   0 |   0 |   937844 |        0 |   0 |        0 |  33607151 |  42 |
|  43 | Henrico        | 69907728 |  2820509 | 39626918 |       0 |  7480192 | 16507646 |  5675782 |   0 |   0 | 11274953 | 27336089 |   0 |  1943537 | 182573354 |  43 |
|  44 | Henry          |  5372178 |  2665097 |  1827121 |       0 |   965378 |   261786 |   231272 |   0 |   0 |   108316 |  2395206 |   0 |        0 |  13826354 |  44 |
|  45 | Highland       |   144402 |    71039 |        0 |       0 |    57765 |    14782 |     8114 |   0 |   0 |    14870 |        0 |   0 |        0 |    310972 |  45 |
|  46 | Isle of Wight  |  2675566 |   971143 |   815733 |       0 |   995345 |    11950 |   571991 |   0 |   0 |    65571 |   502646 |   0 |   131127 |   6741072 |  46 |
|  47 | James City     | 14660594 |        0 |  7187277 |  484998 |   170761 |        0 |  1532445 |   0 |   0 |  2768935 |  6049097 |   0 |   485577 |  33339684 |  47 |
|  48 | King & Queen   |   256207 |   179360 |    27228 |       0 |   207603 |    -2995 |    52013 |   0 |   0 |        0 |        0 |   0 |        0 |    719416 |  48 |
|  49 | King George    |  3191385 |   249808 |  1418654 |       0 |   668937 |   151164 |   433176 |   0 |   0 |   281707 |  1257238 |   0 |   895194 |   8547263 |  49 |
|  50 | King William   |  3416083 |   313265 |   420384 |       0 |   432214 |   134929 |   290488 |   0 |   0 |        0 |   426402 |   0 |        0 |   5433765 |  50 |
|  51 | Lancaster      |  1858276 |        0 |        0 |       0 |   201441 |    38887 |   235218 |   0 |   0 |     1361 |        0 |   0 |    42273 |   2377456 |  51 |
|  52 | Lee            |        0 |        0 |        0 |       0 |        0 |        0 |        0 |   0 |   0 |        0 |        0 |   0 |        0 |         0 |  52 |
|  53 | Loudoun        | 80478625 | 11641488 | 43405100 |       0 |  7296326 |  1955370 | 15579156 |   0 |   0 |  4843313 |        0 |   0 | 21927331 | 187126709 |  53 |
|  54 | Louisa         |  4557042 |   657398 |   257345 |   23717 |  1438729 |        0 |   774923 |   0 |   0 |   189133 |  1208373 |   0 |        0 |   9106660 |  54 |
|  55 | Lunenburg      |   507523 |   184102 |        0 |       0 |   241963 |        0 |    78278 |   0 |   0 |        0 |        0 |   0 |    23791 |   1035657 |  55 |
|  56 | Madison        |  1223259 |   397686 |        0 |   15502 |   472768 |   115133 |   163153 |   0 |   0 |   118565 |   479074 |   0 |        0 |   2985140 |  56 |
|  57 | Mathews        |   633773 |   152689 |   158108 |       0 |   313553 |   107362 |   112837 |   0 |   0 |        0 |   248418 |   0 |    35148 |   1761888 |  57 |
|  58 | Mecklenburg    |  5107176 |   528330 |     1763 |    5701 |   641927 |        0 |   355978 |   0 |   0 |    73228 |        0 |   0 |    72478 |   6786581 |  58 |
|  59 | Middlesex      |  1274255 |   214667 |   355577 |       0 |   285123 |   128552 |   226610 |   0 |   0 |    72466 |   391954 |   0 |    44333 |   2993537 |  59 |
|  60 | Montgomery     | 10021532 |   832200 |        0 |       0 |   805731 |    32371 |  1312098 |   0 |   0 |    45378 |   257089 |   0 |        0 |  13306399 |  60 |
|  61 | Nelson         |  1480484 |   500686 |    35445 |   85728 |   743376 |    94184 |   274289 |   0 |   0 |   500408 |   965975 |   0 |        0 |   4680575 |  61 |
|  62 | New Kent       |  2202411 |   325116 |   894759 |    7845 |   609246 |   106857 |   522868 |   0 |   0 |    19421 |  1127391 |   0 |        0 |   5815914 |  62 |
|  63 | Northampton    |  1443547 |   324208 |    38265 |       0 |   362175 |    28994 |   222204 |   0 |   0 |   475894 |   329119 |   0 |        0 |   3224406 |  63 |
|  64 | Northumberland |   891303 |   360859 |        0 |       0 |   371425 |   297131 |   216908 |   0 |   0 |        0 |        0 |   0 |    54953 |   2192579 |  64 |
|  65 | Nottoway       |  1334885 |   150646 |   219385 |       0 |   197213 |        0 |    63159 |   0 |   0 |     5896 |        0 |   0 |    26899 |   1998083 |  65 |
|  66 | Orange         |  3859456 |   628659 |        0 |   16577 |  1011268 |    91505 |   681011 |   0 |   0 |    46236 |   791214 |   0 |        0 |   7125926 |  66 |
|  67 | Page           |  1864352 |    50731 |   182237 |       0 |   504894 |        0 |   218689 |   0 |   0 |   869855 |   262182 |   0 |        0 |   3952940 |  67 |
|  68 | Patrick        |  1341836 |   410257 |        0 |    4475 |   455497 |    10053 |   131050 |   0 |   0 |   335266 |    42372 |   0 |    54434 |   2785240 |  68 |
|  69 | Pittsylvania   |  3131819 |  1281445 |      600 |   31479 |  2177232 |    79312 |   428615 |   0 |   0 |        0 |   739010 |   0 |   192714 |   8062226 |  69 |
|  70 | Powhatan       |  3845645 |   708455 |    97651 |  277071 |  1171620 |        0 |   587299 |   0 |   0 |    28194 |        0 |   0 |    37222 |   6753157 |  70 |
|  71 | Prince Edward  |  3095612 |   306609 |        0 |   72649 |   579777 |        0 |   183848 |   0 |   0 |        0 |        0 |   0 |        0 |   4238495 |  71 |
|  72 | Prince George  |  3250168 |   905824 |  1701729 |       0 |  1088786 |   119309 |   438652 |   0 |   0 |   574007 |  1188368 |   0 |        0 |   9266843 |  72 |
|  73 | Prince William | 72342000 | 14408000 | 28236000 | 1600000 |  9086000 |  1816000 | 10611000 |   0 |   0 |  4685000 |        0 |   0 |  8216000 | 151000000 |  73 |
|  74 | Pulaski        |  3883192 |   681589 |   947738 |       0 |   597153 |     7955 |   264667 |   0 |   0 |   344864 |  1448275 |   0 |    95356 |   8270789 |  74 |
|  75 | Rappahannock   |   685517 |   174660 |        0 |   30337 |   209143 |    97150 |   163767 |   0 |   0 |    54866 |   219465 |   0 |        0 |   1634905 |  75 |
|  76 | Richmond       |  1585626 |   123087 |        0 |       0 |   256218 |        0 |    84147 |   0 |   0 |        0 |        0 |   0 |    20174 |   2069252 |  76 |
|  78 | Rockbridge     |  3039608 |   601508 |   942041 |       0 |   469104 |   196822 |   285043 |   0 |   0 |  1335612 |  1398932 |   0 |        0 |   8268670 |  78 |
|  79 | Rockingham     |  8047919 |  1733524 |   326346 |       0 |  1342514 |   123430 |  1221494 |   0 |   0 |   827571 |  1187022 |   0 |        0 |  14809820 |  79 |
|  82 | Shenandoah     |  4364517 |   768244 |        0 |   24710 |   916402 |        0 |   463305 |   0 |   0 |   132705 |        0 |   0 |        0 |   6669883 |  82 |
|  83 | Smyth          |  2431494 |   495326 |        0 |       0 |   470778 |        0 |   128947 |   0 |   0 |    61068 |        0 |   0 |   203731 |   3791344 |  83 |
|  84 | Southampton    |   799265 |   523656 |   138742 |       0 |   513327 |    32865 |   164562 |   0 |   0 |     9385 |   165225 |   0 |    68777 |   2415804 |  84 |
|  85 | Spotsylvania   | 20731314 |  2624828 |  5283414 |  732076 |  3292754 |   802914 |  2793094 |   0 |   0 |  1275878 |  9363282 |   0 |  5671360 |  52570914 |  85 |
|  86 | Stafford       | 17540447 |  6597960 |        0 |  283500 |  3000160 |   530217 |  3845119 |   0 |   0 |  1574349 |  8356499 |   0 |  8210254 |  49938505 |  86 |
|  87 | Surry          |   634915 |        0 |   270789 |       0 |   132391 |        0 |    61187 |   0 |   0 |        0 |        0 |   0 |    18863 |   1118145 |  87 |
|  88 | Sussex         |   929808 |    93110 |    29812 |       0 |   222586 |        0 |        0 |   0 |   0 |    48121 |        0 |   0 |    36783 |   1360220 |  88 |
|  90 | Warren         |  4575698 |   741898 |   977580 |       0 |   850319 |        0 |   761318 |   0 |   0 |   210189 |   898042 |   0 |   123134 |   9138178 |  90 |
|  91 | Washington     |  7648906 |  1136827 |        0 |  140019 |  1112830 |    41637 |   541082 |   0 |   0 |   102188 |        0 |   0 |        0 |  10723489 |  91 |
|  92 | Westmoreland   |  1003512 |   340464 |        0 |   51344 |   800075 |        0 |   239358 |   0 |   0 |        0 |        0 |   0 |     5050 |   2439803 |  92 |
|  95 | York           | 15881318 |   224860 |  7527828 |    4713 |  1648997 |   257274 |  1538756 |   0 |   0 |  3757890 |  2997053 |   0 |    10004 |  33848693 |  95 |



labeled_formulas: K=A*J
N=A*J
N=A*K
Q=A+N
K=C*J
N=C*J
N=C*K
K=D*J
N=D*J
N=D*K
K=E*J
J=F*K
N=F*J
N=F*K
K=G*J
J=G*N
K=G*N
K=H*J
J=H*N
K=H*N
K=I*J
N=I*J
N=I*K
K=J*L
K=J*M
J=K*O
K=J*P
K=J*Q
J=L*N
J=N*O
N=J*P
N=J*Q
K=L*N
N=K*O
N=K*P
N=K*Q
K=(A+C)*J
N=(A+C)*J
N=(A+C)*K
N=(A-Q)*C
K=(A+D)*J
N=(A+D)*J
N=(A+D)*K
N=(A-Q)*D
K=(A*E)*J
J=(A*F)*K
N=(A*F)*J
N=(A*F)*K
Q=(F*N)+A
K=(A*G)*J
J=(A*G)*N
K=(A*G)*N
K=(A+H)*J
J=(A*H)*N
K=(A*H)*N
Q=(H*N)+A
K=(A+I)*J
N=(A+I)*J
N=(A+I)*K
N=(A-Q)*I
K=(A*J)*L
K=(A*J)*M
J=(A*K)*O
K=(A*J)*P
J=(A*L)*N
J=(A*N)*O
N=(A*J)*P
K=(A*L)*N
N=(A*K)*O
N=(A*K)*P
Q=(L*N)+A
Q=(N*O)+A
N=(A-Q)*P
K=(C+D)*J
N=(C+D)*J
N=(C+D)*K
K=(C*E)*J
J=(C*F)*K
N=(C*F)*J
N=(C*F)*K
K=(C*G)*J
J=(C*G)*N
K=(C*G)*N
K=(C+H)*J
J=(C*H)*N
K=(C*H)*N
output: {"Arithmetic-Relationship": []}
source: $MMTU_HOME/resources/Relationship-finding/AR/table_2301___xlsx_756f2de2f1b7907d0304e13012e31a03c6e2b186.xlsx___line_310