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 |   S |
|----:|:---------------|--------:|--------:|---------:|-------:|----:|---------:|----:|--------:|---------:|---------:|----:|---------:|--------:|----:|---------:|-----------------:|----:|
|   3 | Alleghany      |       0 |       0 |        0 |      0 |   0 |  1246493 |   0 |       0 |  1246493 |   614687 |   0 |   631806 |       0 |   0 |  1246493 | 474401           |   3 |
|   4 | Amelia         |       0 |       0 |        0 |  13368 |   0 |    53320 |   0 |       0 |    66688 |        0 |   0 |        0 |       0 |   0 |        0 | 567001           |   4 |
|   5 | Amherst        |       0 |   28263 |        0 |  82502 |   0 |        0 |   0 |   21384 |   132149 |  1572683 |   0 |  2112570 |       0 |   0 |  3685253 | 720255           |   5 |
|   6 | Appomattox     |       0 |       0 |        0 |      0 |   0 |    21676 |   0 |       0 |    21676 |        0 |   0 |    52058 |       0 |   0 |    52058 | 141596           |   6 |
|   9 | Bath           |       0 |       0 |        0 |      0 |   0 |   110000 |   0 |       0 |   110000 |   110000 |   0 |        0 |       0 |   0 |   110000 | 201691           |   9 |
|  10 | Bedford        |       0 |       0 |        0 |      0 |   0 |        0 |   0 |       0 |        0 |        0 |   0 |        0 |       0 |   0 |        0 |      0           |  10 |
|  11 | Bland          |       0 |       0 |        0 |      0 |   0 |        0 |   0 |       0 |        0 |        0 |   0 |        0 |       0 |   0 |        0 |  74137.2         |  11 |
|  12 | Botetourt      |       0 |       0 |        0 |      0 |   0 | 12816202 |   0 |       0 | 12816202 | 10926454 |   0 |  1889748 |       0 |   0 | 12816202 | 577601           |  12 |
|  14 | Buchanan       |       0 |  485439 |        0 |      0 |   0 |        0 |   0 |       0 |   485439 |        0 |   0 |  2345892 |       0 |   0 |  2345892 |      2.67229e+07 |  14 |
|  15 | Buckingham     |       0 |       0 |  5150000 |  63120 |   0 |        0 |   0 |       0 |  5213120 |        0 |   0 |  5465578 |       0 |   0 |  5465578 | 913697           |  15 |
|  17 | Caroline       |       0 |       0 |        0 |  39359 |   0 |  2224873 |   0 |   42122 |  2306354 |   108402 |   0 |  1664041 |   41420 |   0 |  1813863 |  89307.3         |  17 |
|  18 | Carroll        |       0 |       0 |        0 |      0 |   0 |        0 |   0 |       0 |        0 |        0 |   0 |        0 |       0 |   0 |        0 | 526056           |  18 |
|  19 | Charles City   |       0 |       0 |        0 |      0 |   0 |   204680 |   0 |       0 |   204680 |        0 |   0 |   397182 |       0 |   0 |   397182 | 395063           |  19 |
|  20 | Charlotte      |       0 |       0 |        0 |   1566 |   0 |        0 |   0 |       0 |     1566 |        0 |   0 |  1627853 |       0 |   0 |  1627853 | 207312           |  20 |
|  25 | Cumberland     |       0 |       0 |        0 |   2042 |   0 |    54047 |   0 |       0 |    56089 |        0 |   0 |        0 |       0 |   0 |        0 | 373725           |  25 |
|  26 | Dickenson      |       0 |       0 |        0 |   6446 |   0 |   839309 |   0 |    6978 |   852733 |        0 |   0 |   839309 |   13424 |   0 |   852733 | -60551.2         |  26 |
|  28 | Essex          |       0 |   22073 |        0 |  15186 |   0 |   371920 |   0 |       0 |   409179 |        0 |   0 |   409179 |       0 |   0 |   409179 | 451039           |  28 |
|  31 | Floyd          |       0 |       0 |  7580935 |      0 |   0 |        0 |   0 |       0 |  7580935 |  3361184 |   0 |    24115 |       0 |   0 |  3385299 |   9051.75        |  31 |
|  32 | Fluvanna       |       0 |       0 |        0 |    777 |   0 |  3875084 |   0 |  112500 |  3988361 |   660646 |   0 |    60985 |       0 |   0 |   721631 | 488433           |  32 |
|  35 | Giles          |       0 |       0 |    24466 |      0 |   0 |   562364 |   0 |       0 |   586830 |        0 |   0 |   586830 |       0 |   0 |   586830 |  23256.2         |  35 |
|  38 | Grayson        |       0 |   16408 |        0 |      0 |   0 |   436936 |   0 |       0 |   453344 |        0 |   0 |   453344 |       0 |   0 |   453344 | 230767           |  38 |
|  39 | Greene         |       0 |       0 |  1040000 |  51673 |   0 |        0 |   0 |       0 |  1091673 |  5061133 |   0 |  6147284 |       0 |   0 | 11208417 |  93276.8         |  39 |
|  44 | Henry          |       0 |  295965 |        0 |      0 |   0 | 23085186 |   0 | 1053565 | 24434716 |  1983386 |   0 | 22517847 |       0 |   0 | 24501233 | 337422           |  44 |
|  45 | Highland       |       0 |       0 |        0 |      0 |   0 |   333935 |   0 |       0 |   333935 |        0 |   0 |        0 |  199808 |   0 |   199808 | 347778           |  45 |
|  46 | Isle of Wight  |       0 | 1780278 |        0 |  81409 |   0 |   563739 |   0 | 1314488 |  3739914 |   186077 |   0 |  2514027 |       0 |   0 |  2700104 |      1.41106e+06 |  46 |
|  48 | King & Queen   |       0 |       0 |        0 |  18922 |   0 |   419085 |   0 |       0 |   438007 |   193474 |   0 |   362055 |    9556 |   0 |   565085 | 143204           |  48 |
|  51 | Lancaster      |       0 |       0 |        0 |  17520 |   0 |   267079 |   0 |       0 |   284599 |        0 |   0 |   137023 | 1967377 |   0 |  2104400 |  18773.4         |  51 |
|  52 | Lee            |       0 |       0 |        0 |      0 |   0 |        0 |   0 |       0 |        0 |        0 |   0 |        0 |       0 |   0 |        0 |      0           |  52 |
|  54 | Louisa         |       0 |       0 |        0 | 294775 |   0 |  2032831 |   0 |       0 |  2327606 |   153512 |   0 |  2174094 |       0 |   0 |  2327606 |      1.07247e+06 |  54 |
|  55 | Lunenburg      |       0 |       0 |        0 |   1090 |   0 |   235170 |   0 |       0 |   236260 |        0 |   0 |   181851 |       0 |   0 |   181851 | 492166           |  55 |
|  56 | Madison        |       0 |       0 |        0 |      0 |   0 |  2868301 |   0 |       0 |  2868301 |  1337455 |   0 |  1530846 |       0 |   0 |  2868301 |  89926.5         |  56 |
|  59 | Middlesex      |       0 |       0 |        0 |  15321 |   0 |        0 |   0 |   50000 |    65321 |   170641 |   0 |   660264 |       0 |   0 |   830905 |  89238.7         |  59 |
|  61 | Nelson         |       0 | 2439506 |        0 |      0 |   0 |  1747608 |   0 |       0 |  4187114 |        0 |   0 |  4187114 |       0 |   0 |  4187114 |      4.74928e+06 |  61 |
|  62 | New Kent       |       0 |       0 |        0 |  60745 |   0 |        0 |   0 |  508638 |   569383 |  3005960 |   0 |  5225093 |       0 |   0 |  8231053 |  29605.2         |  62 |
|  63 | Northampton    |       0 |       0 | 27210036 | 233482 |   0 |        0 |   0 |       0 | 27443518 |        0 |   0 |   127046 |       0 |   0 |   127046 |  28635.5         |  63 |
|  64 | Northumberland |       0 |       0 |        0 |      0 |   0 |   528020 |   0 |   25000 |   553020 |        0 |   0 |   207156 |       0 |   0 |   207156 |    487           |  64 |
|  65 | Nottoway       |       0 |       0 |        0 |   1091 |   0 |        0 |   0 |       0 |     1091 |        0 |   0 |        0 |       0 |   0 |        0 | 453932           |  65 |
|  69 | Pittsylvania   | 1322312 |       0 |        0 |      0 |   0 |   680781 |   0 |       0 |  2003093 |        0 |   0 |  2003093 |       0 |   0 |  2003093 | 664770           |  69 |
|  70 | Powhatan       |       0 |       0 |        0 | 208098 |   0 |        0 |   0 |   17050 |   225148 |  1553003 |   0 |   624476 |       0 |   0 |  2177479 | 518114           |  70 |
|  71 | Prince Edward  |       0 |       0 |        0 |  11931 |   0 |        0 |   0 |  426843 |   438774 |        0 |   0 |  7509744 |       0 |   0 |  7509744 |      2.58003e+06 |  71 |
|  75 | Rappahannock   |       0 |       0 |        0 |      0 |   0 |   336122 |   0 |       0 |   336122 |        0 |   0 |   642495 |       0 |   0 |   642495 | 344761           |  75 |
|  76 | Richmond       |       0 |       0 |   750000 |      0 |   0 |        0 |   0 |       0 |   750000 |        0 |   0 |   737549 |       0 |   0 |   737549 | 178955           |  76 |
|  78 | Rockbridge     |       0 |       0 |        0 |      0 |   0 |   361191 |   0 |       0 |   361191 |        0 |   0 |   356204 |  360770 |   0 |   716974 | 222949           |  78 |
|  80 | Russell        |       0 |       0 |        0 |      0 |   0 |   413535 |   0 |       0 |   413535 |        0 |   0 |   413535 |       0 |   0 |   413535 | 417829           |  80 |
|  81 | Scott          |       0 |       0 |    10134 |      0 |   0 |        0 |   0 |       0 |    10134 |        0 |   0 |   364920 |       0 |   0 |   364920 |      1.61401e+06 |  81 |
|  82 | Shenandoah     |       0 |       0 |        0 | 219316 |   0 |  4080742 |   0 |       0 |  4300058 |        0 |   0 |  9008154 |       0 |   0 |  9008154 |      1.59024e+06 |  82 |
|  83 | Smyth          |       0 |       0 |        0 |      0 |   0 |        0 |   0 |       0 |        0 |        0 |   0 |        0 |       0 |   0 |        0 |  69759           |  83 |
|  87 | Surry          |       0 |  967420 |        0 | 138268 |   0 |  2905744 |   0 |  150000 |  4161432 |   731028 |   0 |  6011315 |       0 |   0 |  6742343 |  10729           |  87 |
|  88 | Sussex         |  523375 |       0 |        0 | 106910 |   0 |  3358108 |   0 |  108697 |  4097090 |        0 |   0 |    10925 |       0 |   0 |    10925 |  46224.3         |  88 |
|  91 | Washington     |       0 |       0 |        0 |      0 |   0 |   831966 |   0 |       0 |   831966 |        0 |   0 |   831966 |       0 |   0 |   831966 |      2.23397e+06 |  91 |



labeled_formulas: I=A*G
M=A*G
P=A*G
S=A+G
M=A*I
P=A*I
S=A+I
P=A*M
S=A+M
G=C*E
I=C*E
P=C*E
I=C*G
G=C*M
P=C*G
I=C*M
I=C*P
P=C*M
G=D*E
I=D*E
M=D*E
P=D*E
I=D*G
G=D*M
G=D*O
I=D*M
I=D*O
P=D*I
P=D*M
P=D*O
G=E*I
G=E*M
P=E*G
I=E*M
P=E*I
P=E*M
I=F*G
G=F*M
P=F*G
I=F*M
P=F*I
P=F*M
I=G*H
M=G*H
P=G*H
I=G*J
I=G*K
I=G*L
I=G*M
I=G*N
S=(C*D)+A
G=(A*C)*E
I=(A*C)*E
P=(A*C)*E
S=(C*E)+A
I=(A*C)*G
G=(A*C)*M
P=(A*C)*G
I=(A*C)*M
I=(A*C)*P
I=(A-S)*C
P=(A*C)*M
S=(C*M)+A
S=(C*O)+A
G=(A*D)*E
I=(A*D)*E
M=(A*D)*E
P=(A*D)*E
S=(D*E)+A
I=(A*D)*G
G=(A*D)*M
G=(A*D)*O
I=(A*D)*M
I=(A*D)*O
P=(A*D)*I
S=(D*I)+A
P=(A*D)*M
S=(D*M)+A
P=(A*D)*O
S=(D*O)+A
G=(A*E)*I
G=(A*E)*M
P=(A*E)*G
I=(A*E)*M
P=(A*E)*I
S=(E*I)+A
S=(E*J)+A
P=(A*E)*M
S=(E*M)+A
S=(E*O)+A
I=(A*F)*G
G=(A*F)*M
P=(A*F)*G
I=(A*F)*M
P=(A*F)*I
S=(F*I)+A
P=(A*F)*M
S=(F*M)+A
I=(A*G)*H
M=(A*G)*H
output: {"Arithmetic-Relationship": []}
source: $MMTU_HOME/resources/Relationship-finding/AR/table_2331___xlsx_756f2de2f1b7907d0304e13012e31a03c6e2b186.xlsx___line_340