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 |
|----:|:---------------|----------:|--------:|----------:|----:|----------:|----------:|----:|----------:|----------:|----------:|----:|----------:|----:|
|   1 | Accomack       |  24796003 |       0 |  71070720 |   0 |  95866723 |  70113096 |   0 |  25753627 |         0 |  95866723 |   0 |  95866723 |   1 |
|   2 | Albemarle      | 168144143 |       0 | 273326447 |   0 | 441470590 | 317112870 |   0 | 124357720 |         0 | 441470590 |   0 | 441470590 |   2 |
|   4 | Amelia         |   3965602 |       0 |  26298252 |   0 |  30263854 |  28022230 |   0 |    680470 |   1561154 |  30263854 |   0 |  30263854 |   4 |
|   5 | Amherst        |  19746083 |       0 |  81597541 |   0 | 101343624 |  68558506 |   0 |  21752295 |  11032823 | 101343624 |   0 | 101343624 |   5 |
|   6 | Appomattox     |  15418336 |       0 |  30045512 |   0 |  45463848 |  28635779 |   0 |  16828069 |         0 |  45463848 |   0 |  45463848 |   6 |
|   9 | Bath           |   2463278 |       0 |  14116734 |   0 |  16580012 |  10929064 |   0 |   4977082 |    673866 |  16580012 |   0 |  16580012 |   9 |
|  10 | Bedford        |         0 |       0 |         0 |   0 |         0 |         0 |   0 |         0 |         0 |         0 |   0 |         0 |  10 |
|  11 | Bland          |  11091882 |       0 |  11679271 |   0 |  22771153 |   6894250 |   0 |   6856685 |   9020218 |  22771153 |   0 |  22771153 |  11 |
|  12 | Botetourt      |  49354147 |  455790 |  65246122 |   0 | 115056059 |  93400541 |   0 |  21655518 |         0 | 115056059 |   0 | 115056059 |  12 |
|  14 | Buchanan       |  11698592 |       0 |  58491821 |   0 |  70190413 |  39414521 |   0 |  17395013 |  13380879 |  70190413 |   0 |  70190413 |  14 |
|  15 | Buckingham     |  30294528 |       0 |  33851552 |   0 |  64146080 |  39798445 |   0 |  13015532 |  11332103 |  64146080 |   0 |  64146080 |  15 |
|  16 | Campbell       |  25468455 | 1591702 | 116103859 |   0 | 143164016 | 110121730 |   0 |  33042286 |         0 | 143164016 |   0 | 143164016 |  16 |
|  17 | Caroline       | 100429582 |       0 |  68375607 |   0 | 168805189 |  90249540 |   0 |  42156863 |  36398786 | 168805189 |   0 | 168805189 |  17 |
|  19 | Charles City   |   3595772 |       0 |  11213478 |   0 |  14809250 |   7902531 |   0 |   5375361 |   1531358 |  14809250 |   0 |  14809250 |  19 |
|  20 | Charlotte      |  18108712 |       0 |  29626627 |   0 |  47735339 |  29070585 |   0 |  18664754 |         0 |  47735339 |   0 |  47735339 |  20 |
|  22 | Clarke         |  24652109 |       0 |  30442992 |   0 |  55095101 |  45049005 |   0 |   6817224 |   3228872 |  55095101 |   0 |  55095101 |  22 |
|  23 | Craig          |   1091602 |       0 |   9233053 |   0 |  10324655 |   8446514 |   0 |   1878141 |         0 |  10324655 |   0 |  10324655 |  23 |
|  26 | Dickenson      |  23006081 |       0 |  47897998 |   0 |  70904079 |  42626056 |   0 |  20394357 |   7883666 |  70904079 |   0 |  70904079 |  26 |
|  27 | Dinwiddie      |  72132893 |       0 |  56633198 |   0 | 128766091 |  81451801 |   0 |  41561102 |   5753188 | 128766091 |   0 | 128766091 |  27 |
|  28 | Essex          |  23106338 |       0 |  15656764 |   0 |  38763102 |  33213751 |   0 |   4747016 |    802335 |  38763102 |   0 |  38763102 |  28 |
|  31 | Floyd          |  20656688 |       0 |  30066904 |   0 |  50723592 |  33476249 |   0 |  17247343 |         0 |  50723592 |   0 |  50723592 |  31 |
|  32 | Fluvanna       |  91277981 |       0 |  47779466 |   0 | 139057447 | 113945467 |   0 |  16457282 |   8654698 | 139057447 |   0 | 139057447 |  32 |
|  33 | Franklin       |  52255755 |       0 | 110808051 |   0 | 163063806 |  91669822 |   0 |  71393984 |         0 | 163063806 |   0 | 163063806 |  33 |
|  34 | Frederick      | 173078192 |       0 | 273583635 |   0 | 446661827 | 324572623 |   0 | 122089204 |         0 | 446661827 |   0 | 446661827 |  34 |
|  36 | Gloucester     |  48078632 |       0 |  81111572 |   0 | 129190204 |  92359565 |   0 |  29478806 |   7351833 | 129190204 |   0 | 129190204 |  36 |
|  37 | Goochland      |  97951795 |       0 |  67840664 |   0 | 165792459 |  34819184 |   0 |  14973115 | 116000160 | 165792459 |   0 | 165792459 |  37 |
|  38 | Grayson        |  13483611 |       0 |  26023235 |   0 |  39506846 |  31700665 |   0 |   7729884 |     76297 |  39506846 |   0 |  39506846 |  38 |
|  41 | Halifax        |  53112561 | 1866791 |  64125317 |   0 | 119104669 |  81711021 |   0 |  37393648 |         0 | 119104669 |   0 | 119104669 |  41 |
|  42 | Hanover        | 161464578 |       0 | 240858617 |   0 | 402323195 | 256617764 |   0 | 126786189 |  18919242 | 402323195 |   0 | 402323195 |  42 |
|  44 | Henry          | 112495219 |       0 |  90512369 |   0 | 203007588 | 100101756 |   0 | 102905832 |         0 | 203007588 |   0 | 203007588 |  44 |
|  45 | Highland       |    124646 |       0 |   5301102 |   0 |   5425748 |   4063352 |   0 |   1112599 |    249797 |   5425748 |   0 |   5425748 |  45 |
|  46 | Isle of Wight  | 139900880 |       0 |  85159507 |   0 | 225060387 | 118384473 |   0 |  70208679 |  36467235 | 225060387 |   0 | 225060387 |  46 |
|  47 | James City     | 161829805 |       0 | 173511655 |   0 | 335341460 | 228036694 |   0 |  74436633 |  32868133 | 335341460 |   0 | 335341460 |  47 |
|  48 | King & Queen   |         0 |       0 |  11508006 |   0 |  11508006 |  10094532 |   0 |   1413474 |         0 |  11508006 |   0 |  11508006 |  48 |
|  49 | King George    |  93651873 |       0 |  54777220 |   0 | 148429093 | 103760247 |   0 |  19689053 |  24979793 | 148429093 |   0 | 148429093 |  49 |
|  50 | King William   |  26570497 |       0 |  27701325 |   0 |  54271822 |  39064670 |   0 |  15207152 |         0 |  54271822 |   0 |  54271822 |  50 |
|  51 | Lancaster      |   6893331 |       0 |  18839702 |   0 |  25733033 |  15916320 |   0 |   9651900 |    164813 |  25733033 |   0 |  25733033 |  51 |
|  52 | Lee            |         0 |       0 |         0 |   0 |         0 |         0 |   0 |         0 |         0 |         0 |   0 |         0 |  52 |
|  54 | Louisa         |  59734388 |       0 |  70249082 |   0 | 129983470 |  78751375 |   0 |  51232095 |         0 | 129983470 |   0 | 129983470 |  54 |
|  55 | Lunenburg      |   8430791 |       0 |  18367797 |   0 |  26798588 |  22093135 |   0 |   4705453 |         0 |  26798588 |   0 |  26798588 |  55 |
|  56 | Madison        |  24641000 |  310000 |  23267818 |   0 |  48218818 |  20370698 |   0 |  27848120 |         0 |  48218818 |   0 |  48218818 |  56 |
|  57 | Mathews        |   1800019 |       0 |  15067525 |   0 |  16867544 |  13486501 |   0 |   3381043 |         0 |  16867544 |   0 |  16867544 |  57 |
|  58 | Mecklenburg    | 127050787 |       0 |  58248700 |   0 | 185299487 | 174614705 |   0 |  10684782 |         0 | 185299487 |   0 | 185299487 |  58 |
|  59 | Middlesex      |  19159100 |       0 |  18020076 |   0 |  37179176 |  27560689 |   0 |   9618487 |         0 |  37179176 |   0 |  37179176 |  59 |
|  61 | Nelson         |  19748849 |       0 |  28902047 |   0 |  48650896 |  35149465 |   0 |  13501431 |         0 |  48650896 |   0 |  48650896 |  61 |
|  62 | New Kent       |  60963753 |       0 |  38751825 |   0 |  99715578 |  66486780 |   0 |  19576362 |  13652436 |  99715578 |   0 |  99715578 |  62 |
|  64 | Northumberland |  30836745 |       0 |  20029729 |   0 |  50866474 |  45536668 |   0 |   3079376 |   2250430 |  50866474 |   0 |  50866474 |  64 |
|  65 | Nottoway       |   1482319 |  155163 |  26670838 |   0 |  28308320 |  22394598 |   0 |   5913722 |         0 |  28308320 |   0 |  28308320 |  65 |
|  66 | Orange         |  76199367 |       0 |  69647323 |   0 | 145846690 |  87840705 |   0 |  57947844 |     58141 | 145846690 |   0 | 145846690 |  66 |
|  67 | Page           |  52329163 |       0 |  59277577 |   0 | 111606740 |  80870027 |   0 |  30736713 |         0 | 111606740 |   0 | 111606740 |  67 |
|  69 | Pittsylvania   |  59301508 |       0 | 120677868 |   0 | 179979376 | 146044699 |   0 |  33765308 |    169369 | 179979376 |   0 | 179979376 |  69 |
|  70 | Powhatan       | 119797062 |       0 |  69747050 |   0 | 189544112 | 134259596 |   0 |  38776764 |  16507752 | 189544112 |   0 | 189544112 |  70 |
|  75 | Rappahannock   |   1244462 |       0 |  18857181 |   0 |  20101643 |  13502155 |   0 |   6599488 |         0 |  20101643 |   0 |  20101643 |  75 |
|  76 | Richmond       |  16864832 |       0 |  15395600 |   0 |  32260432 |  27807053 |   0 |   4453379 |         0 |  32260432 |   0 |  32260432 |  76 |
|  78 | Rockbridge     |  63510241 |  360000 |  59761916 |   0 | 123632157 |  73910262 |   0 |  49721895 |         0 | 123632157 |   0 | 123632157 |  78 |
|  79 | Rockingham     | 129369141 |       0 | 202847909 |   0 | 332217050 | 240693445 |   0 |  76495263 |  15028342 | 332217050 |   0 | 332217050 |  79 |
|  83 | Smyth          |  48010507 |       0 |  57143792 |   0 | 105154299 |  55097088 |   0 |  39640746 |  10416465 | 105154299 |   0 | 105154299 |  83 |
|  87 | Surry          |  20298675 |       0 |  19081596 |   0 |  39380271 |  17403786 |   0 |  21976485 |         0 |  39380271 |   0 |  39380271 |  87 |
|  88 | Sussex         |   6656012 | 4968629 |  19846734 |   0 |  31471375 |  27444134 |   0 |   4027241 |         0 |  31471375 |   0 |  31471375 |  88 |
|  89 | Tazewell       |  50366889 |       0 | 100367987 |   0 | 150734876 |  72123210 |   0 |  40963728 |  37647938 | 150734876 |   0 | 150734876 |  89 |
|  92 | Westmoreland   |  69380502 |       0 |  28455910 |   0 |  97836412 |  67720827 |   0 |  16900082 |  13215503 |  97836412 |   0 |  97836412 |  92 |
|  93 | Wise           |  65942095 |       0 |  93972766 |   0 | 159914861 | 113284561 |   0 |  32514797 |  14115503 | 159914861 |   0 | 159914861 |  93 |



labeled_formulas: I=A*F
M=A*F
M=A*I
O=A+I
O=A+M
I=C*F
M=C*F
M=C*I
F=D*K
M=D*I
I=E*F
M=E*F
M=E*I
I=F*G
M=F*G
N=F+G
I=F*H
M=F*H
I=F*J
F=I*K
I=F*L
F=I*M
I=F*N
I=F*O
M=F*J
F=K*M
M=F*L
N=F+L
M=F*N
M=F*O
L=G+I
M=G*I
N=G+I
M=G-L
M=H*I
M=I*J
M=I*K
M=I*L
N=I+L
M=I*N
M=I*O
I=(A+C)*F
M=(A+C)*F
N=(A/C)+G
M=(A+C)*I
I=(A-O)*C
N=(A/C)+L
M=(A-O)*C
F=(A*D)*K
M=(A*D)*I
O=(D*K)+A
I=(A+E)*F
M=(A+E)*F
N=(A/E)+G
M=(A+E)*I
I=(A-O)*E
N=(A/E)+L
M=(A-O)*E
I=(A*F)*G
M=(A*F)*G
N=(A*F)+G
I=(A*F)*H
M=(A*F)*H
I=(A*F)*J
F=(A*I)*K
I=(A*F)*L
F=(A*I)*M
I=(A*F)*N
M=(A*F)*J
F=(A*K)*M
M=(A*F)*L
N=(A*F)+L
M=(A*F)*N
N=(A/H)+G
L=(A*I)+G
M=(A+G)*I
N=(A*I)+G
I=(A-O)*G
N=(A/J)+G
L=(A*M)+G
N=(A*M)+G
M=(A-O)*G
O=(A+G)-N
M=(A+H)*I
I=(A-O)*H
N=(A/H)+L
M=(A-O)*H
M=(A*I)*J
I=(A-O)*J
M=(A*I)*K
O=(I*K)+A
output: {"Arithmetic-Relationship": []}
source: $MMTU_HOME/resources/Relationship-finding/AR/table_2340___xlsx_756f2de2f1b7907d0304e13012e31a03c6e2b186.xlsx___line_349