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 |
|----:|:-----------------|----:|--------:|-------:|--------:|----:|----:|----:|----:|----:|--------:|----:|----:|----:|----:|
|   2 | ALBEMARLE        |  25 |  650000 |  50000 |  700000 |   0 |   0 |   0 |   0 |   0 |  700000 |   0 |   0 |   0 |   0 |
|   4 | AMELIA           |   3 |   78000 |  50000 |  128000 |   0 |   0 |   0 |   0 |   0 |  128000 |   0 |   0 |   0 |   0 |
|   5 | AMHERST          |   9 |  234000 |  50000 |  284000 |   0 |   0 |   0 |   0 |   0 |  284000 |   0 |   0 |   0 |   0 |
|   6 | APPOMATTOX       |   4 |  104000 |  50000 |  154000 |   0 |   0 |   0 |   0 |   0 |  154000 |   0 |   0 |   0 |   0 |
|   8 | AUGUSTA          |  18 |  468000 |  50000 |  518000 |   0 |   0 |   0 |   0 |   0 |  518000 |   0 |   0 |   0 |   0 |
|  10 | BEDFORD          |  19 |  494000 | 100000 |  594000 |   0 |   0 |   0 |   0 |   0 |  594000 |   0 |   0 |   0 |   0 |
|  12 | BOTETOURT        |  11 |  286000 |  50000 |  336000 |   0 |   0 |   0 |   0 |   0 |  336000 |   0 |   0 |   0 |   0 |
|  13 | BRUNSWICK        |   5 |  130000 |  50000 |  180000 |   0 |   0 |   0 |   0 |   0 |  180000 |   0 |   0 |   0 |   0 |
|  15 | BUCKINGHAM       |   4 |  104000 |  50000 |  154000 |   0 |   0 |   0 |   0 |   0 |  154000 |   0 |   0 |   0 |   0 |
|  16 | CAMPBELL         |  13 |  338000 |  50000 |  388000 |   0 |   0 |   0 |   0 |   0 |  388000 |   0 |   0 |   0 |   0 |
|  17 | CAROLINE         |   5 |  130000 |  50000 |  180000 |   0 |   0 |   0 |   0 |   0 |  180000 |   0 |   0 |   0 |   0 |
|  18 | CARROLL          |   9 |  234000 |  50000 |  284000 |   0 |   0 |   0 |   0 |   0 |  284000 |   0 |   0 |   0 |   0 |
|  19 | CHARLES CITY     |   2 |   52000 |  50000 |  102000 |   0 |   0 |   0 |   0 |   0 |  102000 |   0 |   0 |   0 |   0 |
|  20 | CHARLOTTE        |   5 |  130000 |  50000 |  180000 |   0 |   0 |   0 |   0 |   0 |  180000 |   0 |   0 |   0 |   0 |
|  22 | CLARKE           |   4 |  104000 |  50000 |  154000 |   0 |   0 |   0 |   0 |   0 |  154000 |   0 |   0 |   0 |   0 |
|  23 | CRAIG            |   2 |   52000 |  50000 |  102000 |   0 |   0 |   0 |   0 |   0 |  102000 |   0 |   0 |   0 |   0 |
|  24 | CULPEPER         |  10 |  260000 |  50000 |  310000 |   0 |   0 |   0 |   0 |   0 |  310000 |   0 |   0 |   0 |   0 |
|  25 | CUMBERLAND       |   3 |   78000 |  50000 |  128000 |   0 |   0 |   0 |   0 |   0 |  128000 |   0 |   0 |   0 |   0 |
|  26 | DICKENSON        |   5 |  130000 |  50000 |  180000 |   0 |   0 |   0 |   0 |   0 |  180000 |   0 |   0 |   0 |   0 |
|  27 | DINWIDDIE        |   7 |  182000 |  50000 |  232000 |   0 |   0 |   0 |   0 |   0 |  232000 |   0 |   0 |   0 |   0 |
|  31 | FLOYD            |   5 |  130000 |  50000 |  180000 |   0 |   0 |   0 |   0 |   0 |  180000 |   0 |   0 |   0 |   0 |
|  32 | FLUVANNA         |   5 |  130000 |  50000 |  180000 |   0 |   0 |   0 |   0 |   0 |  180000 |   0 |   0 |   0 |   0 |
|  33 | FRANKLIN         |  16 |  416000 |  50000 |  466000 |   0 |   0 |   0 |   0 |   0 |  466000 |   0 |   0 |   0 |   0 |
|  34 | FREDERICK        |  18 |  468000 |  50000 |  518000 |   0 |   0 |   0 |   0 |   0 |  518000 |   0 |   0 |   0 |   0 |
|  36 | GLOUCESTER       |   8 |  208000 |  50000 |  258000 |   0 |   0 |   0 |   0 |   0 |  258000 |   0 |   0 |   0 |   0 |
|  37 | GOOCHLAND        |   5 |  130000 |  50000 |  180000 |   0 |   0 |   0 |   0 |   0 |  180000 |   0 |   0 |   0 |   0 |
|  38 | GRAYSON          |   7 |  182000 |  50000 |  232000 |   0 |   0 |   0 |   0 |   0 |  232000 |   0 |   0 |   0 |   0 |
|  39 | GREENE           |   5 |  130000 |  50000 |  180000 |   0 |   0 |   0 |   0 |   0 |  180000 |   0 |   0 |   0 |   0 |
|  41 | HALIFAX          |   9 |  234000 |  50000 |  284000 |   0 |   0 |   0 |   0 |   0 |  284000 |   0 |   0 |   0 |   0 |
|  42 | HANOVER          |  23 |  598000 |  50000 |  648000 |   0 |   0 |   0 |   0 |   0 |  648000 |   0 |   0 |   0 |   0 |
|  43 | HENRICO          |  67 | 1742000 |  50000 | 1792000 |   0 |   0 |   0 |   0 |   0 | 1792000 |   0 |   0 |   0 |   0 |
|  44 | HENRY            |  14 |  364000 |  50000 |  414000 |   0 |   0 |   0 |   0 |   0 |  414000 |   0 |   0 |   0 |   0 |
|  45 | HIGHLAND         |   2 |   52000 |  50000 |  102000 |   0 |   0 |   0 |   0 |   0 |  102000 |   0 |   0 |   0 |   0 |
|  46 | ISLE OF WIGHT    |   9 |  234000 |  50000 |  284000 |   0 |   0 |   0 |   0 |   0 |  284000 |   0 |   0 |   0 |   0 |
|  49 | KING QUEEN       |   3 |   78000 |  50000 |  128000 |   0 |   0 |   0 |   0 |   0 |  128000 |   0 |   0 |   0 |   0 |
|  52 | LEE              |  10 |  260000 |  50000 |  310000 |   0 |   0 |   0 |   0 |   0 |  310000 |   0 |   0 |   0 |   0 |
|  53 | LOUDOUN          |  88 | 2288000 |  50000 | 2338000 |   0 |   0 |   0 |   0 |   0 | 2338000 |   0 |   0 |   0 |   0 |
|  54 | LOUISA           |   6 |  156000 |  50000 |  206000 |   0 |   0 |   0 |   0 |   0 |  206000 |   0 |   0 |   0 |   0 |
|  55 | LUNENBURG        |   4 |  104000 |  50000 |  154000 |   0 |   0 |   0 |   0 |   0 |  154000 |   0 |   0 |   0 |   0 |
|  56 | MADISON          |   4 |  104000 |  50000 |  154000 |   0 |   0 |   0 |   0 |   0 |  154000 |   0 |   0 |   0 |   0 |
|  57 | MATHEWS          |   3 |   78000 |  50000 |  128000 |   0 |   0 |   0 |   0 |   0 |  128000 |   0 |   0 |   0 |   0 |
|  58 | MECKLENBURG      |   8 |  208000 |  50000 |  258000 |   0 |   0 |   0 |   0 |   0 |  258000 |   0 |   0 |   0 |   0 |
|  59 | MIDDLESEX        |   3 |   78000 |  50000 |  128000 |   0 |   0 |   0 |   0 |   0 |  128000 |   0 |   0 |   0 |   0 |
|  60 | MONTGOMERY       |  19 |  494000 |  50000 |  544000 |   0 |   0 |   0 |   0 |   0 |  544000 |   0 |   0 |   0 |   0 |
|  62 | NELSON           |   4 |  104000 |  50000 |  154000 |   0 |   0 |   0 |   0 |   0 |  154000 |   0 |   0 |   0 |   0 |
|  65 | NORTHAMPTON      |   4 |  104000 |  50000 |  154000 |   0 |   0 |   0 |   0 |   0 |  154000 |   0 |   0 |   0 |   0 |
|  66 | NORTHUMBERLAND   |   3 |   78000 |  50000 |  128000 |   0 |   0 |   0 |   0 |   0 |  128000 |   0 |   0 |   0 |   0 |
|  67 | NOTTOWAY         |   5 |  130000 |  50000 |  180000 |   0 |   0 |   0 |   0 |   0 |  180000 |   0 |   0 |   0 |   0 |
|  68 | ORANGE           |   9 |  234000 |  50000 |  284000 |   0 |   0 |   0 |   0 |   0 |  284000 |   0 |   0 |   0 |   0 |
|  69 | PAGE             |   8 |  208000 |  50000 |  258000 |   0 |   0 |   0 |   0 |   0 |  258000 |   0 |   0 |   0 |   0 |
|  70 | PATRICK          |   7 |  182000 |  50000 |  232000 |   0 |   0 |   0 |   0 |   0 |  232000 |   0 |   0 |   0 |   0 |
|  71 | PITTSYLVANIA     |  18 |  468000 |  50000 |  518000 |   0 |   0 |   0 |   0 |   0 |  518000 |   0 |   0 |   0 |   0 |
|  72 | POWHATAN         |   6 |  156000 |  50000 |  206000 |   0 |   0 |   0 |   0 |   0 |  206000 |   0 |   0 |   0 |   0 |
|  73 | PRINCE EDWARD    |   3 |   78000 |  50000 |  128000 |   0 |   0 |   0 |   0 |   0 |  128000 |   0 |   0 |   0 |   0 |
|  78 | RAPPAHANNOCK     |   2 |   52000 |  50000 |  102000 |   0 |   0 |   0 |   0 |   0 |  102000 |   0 |   0 |   0 |   0 |
|  79 | RICHMOND         |   2 |   52000 |  50000 |  102000 |   0 |   0 |   0 |   0 |   0 |  102000 |   0 |   0 |   0 |   0 |
|  80 | ROANOKE          |  26 |  676000 |  50000 |  726000 |   0 |   0 |   0 |   0 |   0 |  726000 |   0 |   0 |   0 |   0 |
|  81 | ROCKBRIDGE       |   6 |  156000 |  50000 |  206000 |   0 |   0 |   0 |   0 |   0 |  206000 |   0 |   0 |   0 |   0 |
|  82 | ROCKINGHAM       |  23 |  598000 |  50000 |  648000 |   0 |   0 |   0 |   0 |   0 |  648000 |   0 |   0 |   0 |   0 |
|  83 | RUSSELL          |  11 |  286000 |  50000 |  336000 |   0 |   0 |   0 |   0 |   0 |  336000 |   0 |   0 |   0 |   0 |
|  84 | SCOTT            |  13 |  338000 |  50000 |  388000 |   0 |   0 |   0 |   0 |   0 |  388000 |   0 |   0 |   0 |   0 |
|  85 | SHENANDOAH       |   9 |  234000 |  50000 |  284000 |   0 |   0 |   0 |   0 |   0 |  284000 |   0 |   0 |   0 |   0 |
|  86 | SMYTH            |  13 |  338000 |  50000 |  388000 |   0 |   0 |   0 |   0 |   0 |  388000 |   0 |   0 |   0 |   0 |
|  88 | SPOTSYLVANIA     |  29 |  754000 |  50000 |  804000 |   0 |   0 |   0 |   0 |   0 |  804000 |   0 |   0 |   0 |   0 |
|  89 | STAFFORD         |  30 |  780000 |  50000 |  830000 |   0 |   0 |   0 |   0 |   0 |  830000 |   0 |   0 |   0 |   0 |
|  90 | SURRY            |   3 |   78000 |  50000 |  128000 |   0 |   0 |   0 |   0 |   0 |  128000 |   0 |   0 |   0 |   0 |
|  92 | TAZEWELL         |  15 |  390000 |  50000 |  440000 |   0 |   0 |   0 |   0 |   0 |  440000 |   0 |   0 |   0 |   0 |
|  93 | WARREN           |   9 |  234000 |  50000 |  284000 |   0 |   0 |   0 |   0 |   0 |  284000 |   0 |   0 |   0 |   0 |
|  94 | WASHINGTON       |  15 |  390000 |  50000 |  440000 |   0 |   0 |   0 |   0 |   0 |  440000 |   0 |   0 |   0 |   0 |
|  95 | WESTMORELAND     |   4 |  104000 |  50000 |  154000 |   0 |   0 |   0 |   0 |   0 |  154000 |   0 |   0 |   0 |   0 |
|  96 | WISE             |  11 |  286000 |  50000 |  336000 |   0 |   0 |   0 |   0 |   0 |  336000 |   0 |   0 |   0 |   0 |
|  97 | WYTHE            |  12 |  312000 |  50000 |  362000 |   0 |   0 |   0 |   0 |   0 |  362000 |   0 |   0 |   0 |   0 |
| 102 | BRISTOL          |   6 |  156000 |  50000 |  206000 |   0 |   0 |   0 |   0 |   0 |  206000 |   0 |   0 |   0 |   0 |
| 103 | BUENA VISTA      |   4 |  104000 |  50000 |  154000 |   0 |   0 |   0 |   0 |   0 |  154000 |   0 |   0 |   0 |   0 |
| 104 | CHARLOTTESVILLE  |   9 |  234000 |  50000 |  284000 |   0 |   0 |   0 |   0 |   0 |  284000 |   0 |   0 |   0 |   0 |
| 106 | COLONIAL HEIGHTS |   5 |  130000 |  50000 |  180000 |   0 |   0 |   0 |   0 |   0 |  180000 |   0 |   0 |   0 |   0 |
| 107 | COVINGTON        |   3 |   78000 |  50000 |  128000 |   0 |   0 |   0 |   0 |   0 |  128000 |   0 |   0 |   0 |   0 |
| 109 | FALLS CHURCH     |   4 |  104000 |  50000 |  154000 |   0 |   0 |   0 |   0 |   0 |  154000 |   0 |   0 |   0 |   0 |
| 116 | MARTINSVILLE     |   4 |  104000 |  50000 |  154000 |   0 |   0 |   0 |   0 |   0 |  154000 |   0 |   0 |   0 |   0 |
| 117 | NEWPORT NEWS     |  38 |  988000 |  50000 | 1038000 |   0 |   0 |   0 |   0 |   0 | 1038000 |   0 |   0 |   0 |   0 |
| 118 | NORFOLK          |  45 | 1170000 |  50000 | 1220000 |   0 |   0 |   0 |   0 |   0 | 1220000 |   0 |   0 |   0 |   0 |
| 121 | PORTSMOUTH       |  19 |  494000 |  50000 |  544000 |   0 |   0 |   0 |   0 |   0 |  544000 |   0 |   0 |   0 |   0 |
| 122 | RADFORD          |   4 |  104000 |  50000 |  154000 |   0 |   0 |   0 |   0 |   0 |  154000 |   0 |   0 |   0 |   0 |
| 123 | RICHMOND CITY    |  44 | 1144000 |  50000 | 1194000 |   0 |   0 |   0 |   0 |   0 | 1194000 |   0 |   0 |   0 |   0 |
| 124 | ROANOKE CITY     |  24 |  624000 |  50000 |  674000 |   0 |   0 |   0 |   0 |   0 |  674000 |   0 |   0 |   0 |   0 |
| 126 | STAUNTON         |   5 |  130000 |  50000 |  180000 |   0 |   0 |   0 |   0 |   0 |  180000 |   0 |   0 |   0 |   0 |
| 128 | VIRGINIA BEACH   |  82 | 2132000 |  50000 | 2182000 |   0 |   0 |   0 |   0 |   0 | 2182000 |   0 |   0 |   0 |   0 |
| 131 | WILLIAMSBURG     |  15 |  390000 |  50000 |  440000 |   0 |   0 |   0 |   0 |   0 |  440000 |   0 |   0 |   0 |   0 |
| 135 | FRANKLIN CITY    |   3 |   78000 |  50000 |  128000 |   0 |   0 |   0 |   0 |   0 |  128000 |   0 |   0 |   0 |   0 |
| 136 | CHESAPEAKE CITY  |  45 | 1170000 |  50000 | 1220000 |   0 |   0 |   0 |   0 |   0 | 1220000 |   0 |   0 |   0 |   0 |
| 137 | LEXINGTON        |   2 |   52000 |  50000 |  102000 |   0 |   0 |   0 |   0 |   0 |  102000 |   0 |   0 |   0 |   0 |
| 139 | SALEM            |   6 |  156000 |  50000 |  206000 |   0 |   0 |   0 |   0 |   0 |  206000 |   0 |   0 |   0 |   0 |
| 142 | POQUOSON         |   4 |  104000 |  50000 |  154000 |   0 |   0 |   0 |   0 |   0 |  154000 |   0 |   0 |   0 |   0 |
| 143 | MANASSAS CITY    |   9 |  234000 |  50000 |  284000 |   0 |   0 |   0 |   0 |   0 |  284000 |   0 |   0 |   0 |   0 |
| 144 | MANASSAS PARK    |   4 |  104000 |  50000 |  154000 |   0 |   0 |   0 |   0 |   0 |  154000 |   0 |   0 |   0 |   0 |
| 202 | COLONIAL BEACH   |   2 |   52000 |  50000 |  102000 |   0 |   0 |   0 |   0 |   0 |  102000 |   0 |   0 |   0 |   0 |
| 207 | WEST POINT       |   3 |   78000 |  50000 |  128000 |   0 |   0 |   0 |   0 |   0 |  128000 |   0 |   0 |   0 |   0 |



label: F=D+E
L=F+K
labeled_formulas: F=D+E
L=F+K
H=A*G
I=A*G
J=A*G
K=A*G
O=A*G
P=A*G
I=A*H
J=A*H
K=A*H
O=A*H
P=A*H
J=A*I
K=A*I
O=A*I
P=A*I
K=A*J
O=A*J
P=A*J
O=A*K
P=A*K
N=A*M
P=A*O
H=C*G
I=C*G
J=C*G
K=C*G
O=C*G
P=C*G
I=C*H
J=C*H
K=C*H
O=C*H
P=C*H
J=C*I
K=C*I
O=C*I
P=C*I
K=C*J
O=C*J
P=C*J
O=C*K
P=C*K
N=C*M
P=C*O
L=D+E
H=D*G
I=D*G
J=D*G
K=D*G
O=D*G
H=(A+C)*G
I=(A+C)*G
J=(A+C)*G
K=(A+C)*G
O=(A+C)*G
P=(A+C)*G
I=(A+C)*H
J=(A+C)*H
K=(A+C)*H
O=(A+C)*H
P=(A+C)*H
J=(A+C)*I
K=(A+C)*I
O=(A+C)*I
P=(A+C)*I
K=(A+C)*J
O=(A+C)*J
P=(A+C)*J
O=(A+C)*K
P=(A+C)*K
N=(A+C)*M
P=(A+C)*O
H=(A+D)*G
I=(A+D)*G
J=(A+D)*G
K=(A+D)*G
O=(A+D)*G
P=(A+D)*G
I=(A+D)*H
J=(A+D)*H
K=(A+D)*H
O=(A+D)*H
P=(A+D)*H
J=(A+D)*I
K=(A+D)*I
O=(A+D)*I
P=(A+D)*I
K=(A+D)*J
O=(A+D)*J
P=(A+D)*J
O=(A+D)*K
P=(A+D)*K
N=(A+D)*M
P=(A+D)*O
H=(A+E)*G
I=(A+E)*G
J=(A+E)*G
K=(A+E)*G
O=(A+E)*G
P=(A+E)*G
output: {"Arithmetic-Relationship": ["F=D+E", "L=F+K"]}
source: $MMTU_HOME/resources/Relationship-finding/AR/table_103447___xlsx_e51b674b7d844f32bf8743910c0a2efde1d85856.xlsx___line_450