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 |   R |
|----:|:---------------|--------:|---------:|--------:|--------:|----:|----:|----:|----:|----:|--------:|---------:|--------:|----:|----:|--------:|----:|
|   1 | Accomack       |  115526 |  3763783 |  275095 |  552117 |   0 |   0 |   0 |   0 |   0 |  246962 |  4126271 |  301554 |   0 |   0 |  315534 |   1 |
|   2 | Albemarle      |  760961 | 12170633 |       0 |       0 |   0 |   0 |   0 |   0 |   0 |  909866 | 13841460 |  583162 |   0 |   0 |   59423 |   2 |
|   3 | Alleghany      |   63822 |  1021514 |  403343 |  384756 |   0 |   0 |   0 |   0 |   0 |  158202 |  1243538 |  357654 |   0 |   0 |   54652 |   3 |
|   4 | Amelia         |  174054 |  1698700 |  277452 |  319173 |   0 |   0 |   0 |   0 |   0 |  133968 |  2006722 |  261872 |   0 |   0 |       0 |   4 |
|   5 | Amherst        |  125437 |  2559744 |  337385 |  445460 |   0 |   0 |   0 |   0 |   0 |  238131 |  2923312 |  327524 |   0 |   0 |  108094 |   5 |
|   6 | Appomattox     |  156266 |  1303064 |  317497 |  291973 |   0 |   0 |   0 |   0 |   0 |  177151 |  1636481 |  242533 |   0 |   0 |       0 |   6 |
|   8 | Augusta        |  147707 |  5697173 |  930125 |  554542 |   0 |   0 |   0 |   0 |   0 |  342567 |  6187447 |  495024 |   0 |   0 |  128420 |   8 |
|   9 | Bath           |  187914 |   854311 |  210868 |  264970 |   0 |   0 |   0 |   0 |   0 |  162354 |  1204579 |  191887 |   0 |   0 |   24276 |   9 |
|  10 | Bedford        |       0 |        0 |       0 |       0 |   0 |   0 |   0 |   0 |   0 |       0 |        0 |       0 |   0 |   0 |       0 |  10 |
|  12 | Botetourt      |  254082 |  3274858 |  399118 |  476976 |   0 |   0 |   0 |   0 |   0 |  304584 |  3833524 |  305916 |   0 |   0 |      50 |  12 |
|  13 | Brunswick      |  571469 |  1357478 |  216793 |  231698 |   0 |   0 |   0 |   0 |   0 |  196141 |  2125088 |  264344 |   0 |   0 |       0 |  13 |
|  14 | Buchanan       |  234376 |  1582521 |  382410 |  486931 |   0 |   0 |   0 |   0 |   0 |  266409 |  2083306 |  396063 |   0 |   0 |   10175 |  14 |
|  15 | Buckingham     |  107738 |  1627608 |  265505 |  317351 |   0 |   0 |   0 |   0 |   0 |  198852 |  1934198 |  227386 |   0 |   0 |       0 |  15 |
|  17 | Caroline       |  218670 |  3779547 |  839069 |  653933 |   0 |   0 |   0 |   0 |   0 |  267813 |  4266030 |  299909 |   0 |   0 |    9815 |  17 |
|  18 | Carroll        |  385915 |  1826465 |  396396 |  399918 |   0 |   0 |   0 |   0 |   0 |  239854 |  2452234 |  301329 |   0 |   0 |       0 |  18 |
|  20 | Charlotte      |  143332 |  1153868 |  220450 |  285405 |   0 |   0 |   0 |   0 |   0 |  134801 |  1432001 |  253828 |   0 |   0 |       0 |  20 |
|  22 | Clarke         |   58053 |  2439007 |  206477 |  312875 |   0 |   0 |   0 |   0 |   0 |  137624 |  2634684 |  220229 |   0 |   0 |    5000 |  22 |
|  23 | Craig          |   28915 |   758595 |  163168 |  206773 |   0 |   0 |   0 |   0 |   0 |  123924 |   911434 |  193127 |   0 |   0 |       1 |  23 |
|  24 | Culpeper       |  262826 |  4920506 |  687692 |  638837 |   0 |   0 |   0 |   0 |   0 |  339125 |  5522457 |  350241 |   0 |   0 |       0 |  24 |
|  25 | Cumberland     |   40776 |  1483533 |  251949 |  300006 |   0 |   0 |   0 |   0 |   0 |  129955 |  1654264 |  251030 |   0 |   0 |  403859 |  25 |
|  26 | Dickenson      |  136777 |  1581220 |  396168 |  348011 |   0 |   0 |   0 |   0 |   0 |  222889 |  1940886 |  278657 |   0 |   0 |    4248 |  26 |
|  27 | Dinwiddie      |  111030 |  3042904 |  491547 |  536777 |   0 |   0 |   0 |   0 |   0 |  190141 |  3344075 |  275871 |   0 |   0 |       0 |  27 |
|  28 | Essex          |  389461 |  1841286 |  269254 |  217225 |   0 |   0 |   0 |   0 |   0 |  163120 |  2393867 |  200183 |   0 |   0 |       0 |  28 |
|  31 | Floyd          |  103735 |  1313302 |  265170 |  237544 |   0 |   0 |   0 |   0 |   0 |  181098 |  1598135 |  238028 |   0 |   0 |       0 |  31 |
|  32 | Fluvanna       |  143857 |  2420654 |  376095 |  453123 |   0 |   0 |   0 |   0 |   0 |  219962 |  2784473 |  310935 |   0 |   0 |       0 |  32 |
|  33 | Franklin       |  452991 |  4432647 |  604553 |  547884 |   0 |   0 |   0 |   0 |   0 |  321492 |  5207130 |  595907 |   0 |   0 |       0 |  33 |
|  34 | Frederick      | 2486524 |  9211466 | 1595848 | 1533359 |   0 |   0 |   0 |   0 |   0 |  167842 | 11865832 |  505563 |   0 |   0 |       0 |  34 |
|  35 | Giles          |  152752 |  1674814 |  329502 |  476584 |   0 |   0 |   0 |   0 |   0 |  192340 |  2019906 |  277146 |   0 |   0 |   58696 |  35 |
|  36 | Gloucester     |  162367 |  6021680 |  528465 |  712373 |   0 |   0 |   0 |   0 |   0 |  248986 |  6433033 |  339290 |   0 |   0 |       0 |  36 |
|  38 | Grayson        |   46790 |  1362846 |  257697 |  365409 |   0 |   0 |   0 |   0 |   0 |  159153 |  1568789 |  234844 |   0 |   0 |   57495 |  38 |
|  39 | Greene         |   92196 |  1350209 |  233951 |  407688 |   0 |   0 |   0 |   0 |   0 |  144431 |  1586836 |  218155 |   0 |   0 |   59283 |  39 |
|  40 | Greensville    |  165998 |  2037251 |  304239 |  316637 |   0 |   0 |   0 |   0 |   0 |  121242 |  2324491 |  256768 |   0 |   0 |   52202 |  40 |
|  41 | Halifax        |  209256 |  2020836 |  415009 |  459656 |   0 |   0 |   0 |   0 |   0 |  225752 |  2455844 |  343494 |   0 |   0 |   12079 |  41 |
|  43 | Henrico        | 1213616 | 50649839 |       0 |       0 |   0 |   0 |   0 |   0 |   0 | 1928898 | 53792353 | 1133909 |   0 |   0 | 1043335 |  43 |
|  45 | Highland       |   31904 |   652924 |  182988 |  180156 |   0 |   0 |   0 |   0 |   0 |   90950 |   775778 |  198286 |   0 |   0 |     184 |  45 |
|  46 | Isle of Wight  |  326003 |  3646120 |  676354 |  790796 |   0 |   0 |   0 |   0 |   0 |  285721 |  4257844 |  343722 |   0 |   0 |   90315 |  46 |
|  47 | James City     |  160125 |  8355122 | 1094759 | 1430684 |   0 |   0 |   0 |   0 |   0 |  470616 |  8985863 |  551587 |   0 |   0 |   63771 |  47 |
|  48 | King & Queen   |   69913 |  1341609 |  216311 |  243542 |   0 |   0 |   0 |   0 |   0 |  133114 |  1544636 |  204287 |   0 |   0 |       0 |  48 |
|  49 | King George    |  112968 |  3296555 |  444132 |  375204 |   0 |   0 |   0 |   0 |   0 |  240719 |  3650242 |  273254 |   0 |   0 |       0 |  49 |
|  51 | Lancaster      |   49727 |  1161890 |  269617 |  310935 |   0 |   0 |   0 |   0 |   0 |  189185 |  1400802 |  231538 |   0 |   0 |       0 |  51 |
|  52 | Lee            |       0 |        0 |       0 |       0 |   0 |   0 |   0 |   0 |   0 |       0 |        0 |       0 |   0 |   0 |       0 |  52 |
|  53 | Loudoun        | 4324859 | 66410096 | 8506112 | 5831587 |   0 |   0 |   0 |   0 |   0 | 2317196 | 73052151 |  991035 |   0 |   0 |  988630 |  53 |
|  55 | Lunenburg      |   48835 |   969620 |  214154 |  223600 |   0 |   0 |   0 |   0 |   0 |  137251 |  1155706 |  229760 |   0 |   0 |       0 |  55 |
|  57 | Mathews        |  119712 |  1649055 |  241198 |  285839 |   0 |   0 |   0 |   0 |   0 |  115192 |  1883959 |  212657 |   0 |   0 |       0 |  57 |
|  58 | Mecklenburg    |  240684 |  3204283 |  698483 |  681881 |   0 |   0 |   0 |   0 |   0 |  248853 |  3693820 |  311462 |   0 |   0 |     157 |  58 |
|  59 | Middlesex      |  100851 |  1450804 |  246122 |  266747 |   0 |   0 |   0 |   0 |   0 |  175537 |  1727192 |  240699 |   0 |   0 |    3465 |  59 |
|  60 | Montgomery     |  238631 |  7108812 | 1061415 |  937675 |   0 |   0 |   0 |   0 |   0 |  497381 |  7844824 |  465256 |   0 |   0 |      84 |  60 |
|  61 | Nelson         |  129128 |  2264182 |  255105 |  340407 |   0 |   0 |   0 |   0 |   0 |  205817 |  2599127 |  258498 |   0 |   0 |    3216 |  61 |
|  62 | New Kent       |  155207 |  4040328 |  674352 |  448431 |   0 |   0 |   0 |   0 |   0 |  247292 |  4442827 |  254054 |   0 |   0 |     771 |  62 |
|  64 | Northumberland |  281701 |  1414697 |  334489 |  367505 |   0 |   0 |   0 |   0 |   0 |  153964 |  1850362 |  264664 |   0 |   0 |       0 |  64 |
|  65 | Nottoway       |  416235 |   519951 |  190339 |  227529 |   0 |   0 |   0 |   0 |   0 |  145463 |  1081649 |  216130 |   0 |   0 |       0 |  65 |
|  66 | Orange         |   97610 |  2125226 |  567144 |  394378 |   0 |   0 |   0 |   0 |   0 |   95129 |  2317965 |  278485 |   0 |   0 |       0 |  66 |
|  67 | Page           |  258539 |  2651367 |  883540 |  593728 |   0 |   0 |   0 |   0 |   0 |  168755 |  3078661 |  310115 |   0 |   0 |   38323 |  67 |
|  68 | Patrick        |   65062 |  1490682 |  284363 |  336455 |   0 |   0 |   0 |   0 |   0 |  134494 |  1690238 |  251005 |   0 |   0 |   14490 |  68 |
|  69 | Pittsylvania   |  209269 |  2759717 |  639944 |  708746 |   0 |   0 |   0 |   0 |   0 |  300464 |  3269450 |  443527 |   0 |   0 |   30649 |  69 |
|  70 | Powhatan       |   83780 |  3858905 |  555524 |  602150 |   0 |   0 |   0 |   0 |   0 |  227485 |  4170170 |  303425 |   0 |   0 |       0 |  70 |
|  71 | Prince Edward  |  118320 |  1438103 |  293512 |  393078 |   0 |   0 |   0 |   0 |   0 |  172024 |  1728447 |  247630 |   0 |   0 |    6291 |  71 |
|  72 | Prince George  |  197523 |  4732738 |  459891 |  601221 |   0 |   0 |   0 |   0 |   0 |  262253 |  5192514 |  470702 |   0 |   0 |  326285 |  72 |
|  73 | Prince William | 3821000 | 38932000 |       0 |       0 |   0 |   0 |   0 |   0 |   0 | 2513000 | 45266000 |  718000 |   0 |   0 |  597000 |  73 |
|  74 | Pulaski        |  202245 |  2205674 |  472721 |  550519 |   0 |   0 |   0 |   0 |   0 |  236297 |  2644216 |  327656 |   0 |   0 |    1395 |  74 |
|  75 | Rappahannock   |  248343 |  1123300 |  258493 |  311389 |   0 |   0 |   0 |   0 |   0 |  217348 |  1588991 |  192367 |   0 |   0 |    1200 |  75 |
|  77 | Roanoke        |  363038 | 12017561 |  881186 | 1006997 |   0 |   0 |   0 |   0 |   0 |  583367 | 12963966 |  658448 |   0 |   0 |  691192 |  77 |
|  78 | Rockbridge     |  130085 |  1643151 |  327768 |  333759 |   0 |   0 |   0 |   0 |   0 |  191738 |  1964974 |  270054 |   0 |   0 |       0 |  78 |
|  79 | Rockingham     |  185423 |  5312008 |  871418 |  761948 |   0 |   0 |   0 |   0 |   0 |  394225 |  5891656 |  453588 |   0 |   0 |  551529 |  79 |
|  80 | Russell        |  237865 |  1476931 |  334440 |  427604 |   0 |   0 |   0 |   0 |   0 |  228971 |  1943767 |  350529 |   0 |   0 |       0 |  80 |
|  81 | Scott          |  495343 |   965172 |  213945 |  237278 |   0 |   0 |   0 |   0 |   0 |  183052 |  1643567 |  274863 |   0 |   0 |       0 |  81 |
|  82 | Shenandoah     |  237433 |  2158491 |  481175 |  607224 |   0 |   0 |   0 |   0 |   0 |  273917 |  2669841 |  345199 |   0 |   0 |   75382 |  82 |
|  83 | Smyth          |  486956 |  1579955 |  299753 |  359953 |   0 |   0 |   0 |   0 |   0 |  184581 |  2251492 |  345060 |   0 |   0 |    7362 |  83 |
|  84 | Southampton    |  277747 |  1384158 |  500223 |  425039 |   0 |   0 |   0 |   0 |   0 |  211099 |  1873004 |  296960 |   0 |   0 |  306103 |  84 |
|  86 | Stafford       |  904321 | 14719803 | 2963439 | 2196397 |   0 |   0 |   0 |   0 |   0 |  619689 | 16243813 |  642336 |   0 |   0 |       0 |  86 |
|  87 | Surry          |  116398 |  1614272 |  229027 |  265227 |   0 |   0 |   0 |   0 |   0 |  136426 |  1867096 |  201742 |   0 |   0 |     186 |  87 |
|  88 | Sussex         |  160066 |  1840551 |  248810 |  426163 |   0 |   0 |   0 |   0 |   0 |  182730 |  2183347 |  209559 |   0 |   0 |       0 |  88 |
|  89 | Tazewell       |  121813 |  2455941 |  529642 |  647524 |   0 |   0 |   0 |   0 |   0 |  331810 |  2909564 |  413532 |   0 |   0 |  219256 |  89 |
|  90 | Warren         | 2404635 |  2820418 |  683401 |  499060 |   0 |   0 |   0 |   0 |   0 |  262496 |  5487549 |  347387 |   0 |   0 |  188406 |  90 |
|  91 | Washington     |  259237 |  3343283 |  680311 |  861575 |   0 |   0 |   0 |   0 |   0 |  315751 |  3918271 |  420656 |   0 |   0 |  456133 |  91 |
|  94 | Wythe          |  255596 |  1174130 |  285186 |  353315 |   0 |   0 |   0 |   0 |   0 |  135145 |  1564871 |  322966 |   0 |   0 |       0 |  94 |



label: M=(C+D+L)
labeled_formulas: O=A*G
P=A*G
O=A*H
P=A*H
O=A*I
P=A*I
O=A*J
O=A*K
P=A*K
P=A*O
R=A+O
O=C*G
P=C*G
O=C*H
P=C*H
O=C*I
P=C*I
O=C*J
O=C*K
P=C*K
P=C*O
O=D*G
P=D*G
O=D*H
P=D*H
O=D*I
P=D*I
O=D*J
O=D*K
P=D*K
P=D*O
G=E*O
P=E*G
H=E*O
P=E*H
I=E*O
P=E*I
J=E*O
K=E*O
P=E*K
P=E*O
G=F*O
P=F*G
H=F*O
P=F*H
I=F*O
P=F*I
J=F*O
K=F*O
P=F*K
M=(C+D+L)
O=(A+C)*G
P=(A+C)*G
O=(A+C)*H
P=(A+C)*H
O=(A+C)*I
P=(A+C)*I
O=(A+C)*J
O=(A+C)*K
P=(A+C)*K
P=(A+C)*O
O=(A-R)*C
O=(A+D)*G
P=(A+D)*G
O=(A+D)*H
P=(A+D)*H
O=(A+D)*I
P=(A+D)*I
O=(A+D)*J
O=(A+D)*K
P=(A+D)*K
P=(A+D)*O
O=(A-R)*D
G=(A*E)*O
P=(A+E)*G
H=(A*E)*O
P=(A+E)*H
I=(A*E)*O
P=(A+E)*I
J=(A*E)*O
K=(A*E)*O
P=(A+E)*K
P=(A*E)*O
R=(E*O)+A
G=(A*F)*O
P=(A+F)*G
H=(A*F)*O
P=(A+F)*H
I=(A*F)*O
P=(A+F)*I
J=(A*F)*O
K=(A*F)*O
P=(A+F)*K
P=(A*F)*O
R=(F*O)+A
O=(A*G)*L
P=(A*G)*L
O=(A*G)*M
P=(A*G)*M
O=(A*G)*N
P=(A*G)*N
output: {"Arithmetic-Relationship": ["M=(C+D+L)"]}
source: $MMTU_HOME/resources/Relationship-finding/AR/table_2307___xlsx_756f2de2f1b7907d0304e13012e31a03c6e2b186.xlsx___line_316