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 |
|:----------------------|----:|----:|----:|----:|----:|----:|----:|----:|----:|----:|----:|----:|----:|----:|----:|----:|----:|
| ACCOMACK              |  14 |   2 |  28 |   1 |   5 |   0 |   0 |   0 |   0 |   0 |   0 |   4 |  20 |   2 |  23 |   0 |  99 |
| ALBEMARLE             |  30 |   1 |  35 |   0 |   6 |   0 |   0 |   0 |   0 |  10 |   0 |   8 | 162 |  10 |  38 |   7 | 307 |
| ALLEGHANY             |   3 |   1 |   4 |   0 |   1 |   0 |   1 |   0 |   0 |   1 |   0 |   1 |  12 |   1 |   4 |   0 |  29 |
| AMELIA                |   9 |   0 |   6 |   0 |   2 |   0 |   0 |   1 |   0 |   0 |   0 |   1 |   6 |   3 |   5 |   0 |  33 |
| APPOMATTOX            |   5 |   0 |   5 |   0 |   1 |   0 |   0 |   1 |   0 |   1 |   0 |   0 |  12 |   3 |   3 |   1 |  32 |
| ARLINGTON             |  11 |   0 |  44 |   0 |   0 |   0 |   1 |   0 |   0 |  30 |   1 |   8 | 392 |   8 | 113 |  22 | 630 |
| AUGUSTA               |  28 |   6 |  41 |   5 |  12 |   0 |   0 |   0 |   0 |   1 |   2 |  11 |  99 |  10 |  37 |   1 | 253 |
| BATH                  |   1 |   0 |   1 |   0 |   3 |   0 |   0 |   0 |   0 |   1 |   0 |   2 |   9 |   0 |   4 |   0 |  21 |
| BEDFORD CITY          |   1 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   1 |   1 |   0 |   3 |
| BLAND                 |   2 |   1 |   5 |   0 |   1 |   0 |   0 |   0 |   0 |   0 |   0 |   1 |   5 |   7 |   3 |   0 |  25 |
| BOTETOURT             |  15 |   0 |  24 |   1 |   2 |   0 |   0 |   1 |   0 |   0 |   0 |   6 |  47 |   4 |  19 |   1 | 120 |
| BRUNSWICK             |   5 |   0 |  10 |   0 |   1 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |  10 |   1 |   0 |   0 |  27 |
| BUCHANAN              |  10 |   1 |   8 |   0 |   3 |   0 |   0 |   0 |   0 |   0 |   0 |   4 |  15 |   4 |   4 |   1 |  50 |
| BUCKINGHAM            |   4 |   0 |   2 |   0 |   1 |   1 |   0 |   0 |   0 |   0 |   0 |   0 |   1 |   1 |   5 |   0 |  15 |
| BUENA VISTA CITY      |   3 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   3 |   0 |   2 |   0 |   8 |
| CAROLINE              |   7 |   1 |  14 |   0 |   1 |   0 |   0 |   0 |   0 |   0 |   0 |   6 |  24 |   4 |  11 |   0 |  68 |
| CARROLL               |  15 |   1 |  14 |   0 |   2 |   0 |   0 |   0 |   0 |   0 |   1 |   4 |  25 |   5 |   8 |   0 |  75 |
| CHARLES CITY COUNTY   |   4 |   1 |  13 |   0 |   0 |   0 |   0 |   1 |   0 |   0 |   0 |   0 |   3 |   1 |   4 |   2 |  29 |
| CHARLOTTE             |   4 |   0 |   1 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   7 |   2 |   3 |   0 |  17 |
| CHARLOTTESVILLE CITY  |   9 |   0 |   9 |   0 |   0 |   0 |   1 |   0 |   0 |   1 |   0 |   9 |  58 |   0 |  13 |   3 | 103 |
| CLARKE                |   6 |   1 |  10 |   0 |   6 |   0 |   0 |   0 |   0 |   1 |   0 |   1 |  23 |   7 |  11 |   0 |  66 |
| CLIFTON FORGE CITY    |   0 |   0 |   0 |   0 |   1 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   1 |
| COLONIAL HEIGHTS CITY |   3 |   0 |   4 |   0 |   1 |   0 |   0 |   0 |   0 |   0 |   0 |   2 |  20 |   2 |   6 |   0 |  38 |
| COVINGTON CITY        |   5 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   7 |   1 |   1 |   0 |  14 |
| CRAIG                 |   0 |   0 |   2 |   0 |   1 |   0 |   0 |   0 |   0 |   0 |   1 |   0 |   4 |   1 |   0 |   0 |   9 |
| CULPEPER              |  36 |   4 |  40 |   9 |   4 |   0 |   0 |   1 |   0 |   1 |   0 |   3 |  43 |  10 |  29 |   0 | 180 |
| CUMBERLAND            |   1 |   1 |   4 |   0 |   1 |   0 |   0 |   0 |   0 |   1 |   0 |   0 |   4 |   1 |   1 |   0 |  14 |
| DANVILLE CITY         |   8 |   2 |  15 |   0 |   7 |   0 |   0 |   0 |   0 |   0 |   0 |   1 |  28 |   3 |   9 |   0 |  73 |
| DICKENSON             |  10 |   0 |   6 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   2 |   7 |   2 |   2 |   0 |  29 |
| DINWIDDIE             |  22 |   5 |  15 |   1 |   1 |   0 |   0 |   0 |   0 |   0 |   0 |   2 |  18 |   1 |   9 |   0 |  74 |
| EMPORIA CITY          |   1 |   0 |   4 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   1 |   0 |   0 |   2 |   0 |   8 |
| ESSEX                 |   8 |   0 |   6 |   0 |   6 |   0 |   0 |   0 |   0 |   0 |   0 |   1 |   2 |   0 |   7 |   0 |  30 |
| FAIRFAX CITY          |   4 |   2 |  19 |   1 |   1 |   0 |   0 |   0 |   0 |   6 |   0 |   3 |  81 |   4 |  29 |   1 | 151 |
| FALLS CHURCH CITY     |   3 |   0 |   9 |   0 |   2 |   0 |   0 |   0 |   0 |   4 |   0 |   1 |  50 |   2 |   9 |   4 |  84 |
| FRANKLIN CITY         |   4 |   0 |   4 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   7 |   0 |   0 |   0 |  15 |
| FREDERICKSBURG CITY   |  11 |   0 |  19 |   1 |   2 |   1 |   0 |   0 |   0 |   2 |   0 |   5 |  38 |   4 |  11 |   5 |  99 |
| GALAX                 |   5 |   1 |   5 |   0 |   2 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   8 |   1 |   3 |   0 |  25 |
| GILES                 |   6 |   2 |  10 |   0 |   1 |   0 |   0 |   0 |   0 |   1 |   2 |   3 |  19 |   2 |   8 |   0 |  54 |
| GLOUCESTER            |  18 |   0 |  15 |   0 |   6 |   0 |   0 |   0 |   0 |   1 |   0 |   2 |  29 |   9 |  14 |   2 |  96 |
| GOOCHLAND             |   9 |   0 |  20 |   0 |   6 |   0 |   0 |   0 |   0 |   3 |   0 |   3 |  53 |   2 |  18 |   2 | 116 |
| GRAYSON               |  10 |   0 |   2 |   0 |   1 |   0 |   0 |   0 |   0 |   0 |   0 |   2 |   4 |   4 |   3 |   0 |  26 |
| GREENE                |  13 |   1 |   9 |   0 |   0 |   0 |   0 |   1 |   0 |   4 |   0 |   2 |  21 |   2 |   4 |   0 |  57 |
| GREENSVILLE           |   2 |   0 |   4 |   0 |   1 |   0 |   0 |   0 |   0 |   0 |   0 |   2 |   0 |   0 |   0 |   0 |   9 |
| HALIFAX               |   6 |   3 |  17 |   1 |   4 |   0 |   0 |   2 |   0 |   0 |   0 |   6 |  10 |   5 |  15 |   0 |  69 |
| HARRISONBURG CITY     |  19 |   0 |  11 |   5 |   6 |   0 |   0 |   1 |   0 |   1 |   0 |   3 |  31 |   7 |  15 |   1 | 100 |
| HENRY                 |  20 |  13 |  15 |   0 |   5 |   0 |   0 |   0 |   0 |   0 |   2 |   8 |  45 |   3 |  13 |   1 | 125 |
| HIGHLAND              |   0 |   0 |   2 |   0 |   0 |   0 |   0 |   0 |   0 |   1 |   0 |   0 |   6 |   0 |   2 |   0 |  11 |
| HOPEWELL CITY         |   4 |   1 |   2 |   0 |   2 |   0 |   0 |   0 |   0 |   0 |   0 |   1 |  11 |   0 |   2 |   0 |  23 |
| ISLE OF WIGHT         |  19 |   0 |  17 |   0 |   8 |   0 |   0 |   0 |   0 |   1 |   2 |   5 |  41 |   3 |  15 |   1 | 112 |
| JAMES CITY COUNTY     |  13 |   0 |  33 |   0 |   6 |   0 |   0 |   1 |   0 |   5 |   0 |  10 | 108 |  19 |  42 |   2 | 239 |
| KING & QUEEN          |   1 |   0 |   5 |   0 |   5 |   0 |   0 |   0 |   0 |   0 |   0 |   1 |   4 |   1 |   0 |   0 |  17 |
| KING GEORGE           |  19 |   1 |  20 |   0 |   4 |   0 |   0 |   0 |   0 |   1 |   1 |   1 |  35 |   6 |  13 |   0 | 101 |
| LANCASTER             |   7 |   0 |  10 |   1 |   2 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |  16 |   0 |   5 |   0 |  41 |
| LEE                   |  13 |   1 |   7 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   2 |  19 |   2 |   5 |   0 |  49 |
| LEXINGTON CITY        |   0 |   0 |   3 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   8 |   0 |   1 |   0 |  12 |
| LOUISA                |  10 |   1 |  25 |   0 |  11 |   0 |   0 |   0 |   0 |   3 |   0 |   2 |  27 |   7 |  16 |   0 | 102 |
| LUNENBURG             |   2 |   1 |   5 |   0 |   2 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   6 |   2 |   0 |   1 |  19 |
| MADISON               |   8 |   0 |  11 |   0 |   1 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |  10 |   1 |   8 |   0 |  39 |
| MANASSAS PARK CITY    |   2 |   0 |  13 |   0 |   1 |   0 |   0 |   0 |   0 |   0 |   1 |   2 |  12 |   1 |   2 |   0 |  34 |
| MARTINSVILLE CITY     |   2 |   0 |   1 |   0 |   1 |   0 |   0 |   0 |   0 |   1 |   0 |   3 |   7 |   0 |   4 |   0 |  19 |
| MATHEWS               |   5 |   0 |   6 |   0 |   2 |   0 |   0 |   0 |   0 |   1 |   0 |   3 |   5 |   0 |   4 |   0 |  26 |
| MECKLENBURG           |  11 |   2 |  21 |   1 |   9 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |  21 |   4 |   6 |   0 |  75 |
| MILITARY              |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   1 |   1 |   0 |   0 |   0 |   2 |
| NELSON                |   3 |   0 |   6 |   0 |   5 |   0 |   0 |   0 |   0 |   0 |   1 |   0 |  15 |   2 |  10 |   0 |  42 |
| NEW KENT              |  16 |   0 |  24 |   0 |   4 |   0 |   0 |   1 |   0 |   1 |   0 |   4 |  36 |   3 |  10 |   0 |  99 |
| NEWPORT NEWS CITY     |  79 |  25 | 146 |   0 |  12 |   0 |   1 |   0 |   0 |   5 |   3 |  37 | 332 |  44 |  68 |   2 | 754 |
| NORTHAMPTON           |  11 |   0 |   7 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   2 |   9 |   0 |   4 |   0 |  33 |
| NORTON CITY           |   2 |   0 |   1 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   3 |   0 |   0 |   0 |   6 |
| PAGE                  |   4 |   3 |   9 |   0 |   2 |   0 |   0 |   0 |   0 |   0 |   0 |   1 |  15 |   0 |   6 |   0 |  40 |
| PATRICK               |   9 |   2 |   9 |   0 |   3 |   0 |   0 |   0 |   0 |   1 |   1 |   4 |  12 |   2 |   6 |   0 |  49 |
| PETERSBURG CITY       |  14 |   2 |   5 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   4 |  19 |   1 |   7 |   0 |  52 |
| PITTSYLVANIA          |  22 |   1 |  25 |   2 |   6 |   0 |   0 |   2 |   0 |   0 |   0 |  12 |  46 |   9 |  17 |   1 | 143 |
| POQUOSON CITY         |   5 |   0 |  10 |   0 |   1 |   0 |   0 |   0 |   0 |   1 |   0 |   2 |  15 |   4 |   7 |   1 |  46 |
| POWHATAN              |  15 |   0 |  24 |   0 |   9 |   0 |   0 |   1 |   0 |   1 |   0 |   4 |  32 |   6 |  13 |   2 | 107 |
| PRINCE EDWARD         |   3 |   0 |   5 |   0 |   2 |   0 |   0 |   0 |   0 |   0 |   0 |   1 |   6 |   0 |  12 |   2 |  31 |
| PULASKI               |  18 |   2 |  16 |   3 |   1 |   0 |   0 |   0 |   0 |   1 |   2 |   2 |  17 |   4 |  11 |   1 |  78 |
| RADFORD CITY          |   9 |   2 |   3 |   1 |   0 |   0 |   0 |   0 |   0 |   2 |   2 |   0 |  11 |   0 |   3 |   0 |  33 |
| RAPPAHANNOCK          |   2 |   0 |   8 |   0 |   0 |   0 |   0 |   0 |   0 |   1 |   0 |   0 |   6 |   1 |   8 |   0 |  26 |
| RICHMOND COUNTY       |   5 |   0 |   1 |   0 |   3 |   0 |   0 |   0 |   0 |   0 |   0 |   1 |  10 |   1 |   4 |   0 |  25 |
| ROCKBRIDGE            |   0 |   0 |  12 |   0 |   3 |   0 |   0 |   0 |   0 |   0 |   0 |   8 |  20 |   3 |  10 |   0 |  56 |
| SALEM CITY            |  12 |   0 |  20 |   1 |   2 |   0 |   0 |   1 |   0 |   0 |   1 |   2 |  33 |   5 |   6 |   0 |  83 |
| SCOTT                 |   3 |   1 |   3 |   1 |   2 |   0 |   0 |   0 |   0 |   0 |   0 |   3 |   8 |   5 |   6 |   0 |  32 |
| SHENANDOAH            |  14 |   1 |  25 |   0 |   5 |   0 |   0 |   1 |   0 |   0 |   0 |   2 |  42 |   6 |   7 |   9 | 112 |
| SMYTH                 |  12 |   3 |   6 |   0 |   4 |   0 |   0 |   0 |   0 |   0 |   0 |   2 |  19 |   2 |   8 |   0 |  56 |
| SOUTH BOSTON CITY     |   2 |   0 |   0 |   1 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   1 |   1 |   0 |   0 |   0 |   5 |
| SOUTHAMPTON           |  11 |   0 |  10 |   0 |   3 |   0 |   0 |   0 |   0 |   0 |   1 |   1 |   9 |   1 |   4 |   0 |  40 |
| STAFFORD              |  58 |   8 |  89 |   3 |  12 |   0 |   0 |   0 |   0 |  10 |   3 |  16 | 217 |  31 |  92 |   3 | 542 |
| STAUNTON CITY         |   5 |   2 |   8 |   0 |   6 |   0 |   0 |   0 |   0 |   1 |   1 |   2 |  38 |   6 |   6 |   0 |  75 |
| SURRY                 |   2 |   1 |   4 |   0 |   1 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   3 |   0 |   4 |   0 |  15 |
| SUSSEX                |   3 |   0 |   2 |   0 |   4 |   0 |   0 |   0 |   0 |   0 |   0 |   1 |   2 |   0 |   3 |   0 |  15 |
| TAZEWELL              |  23 |   2 |  23 |   0 |   7 |   0 |   0 |   0 |   0 |   0 |   0 |   6 |  44 |   8 |  11 |   0 | 124 |
| WARREN                |  29 |   2 |  13 |   0 |   9 |   0 |   0 |   0 |   0 |   2 |   0 |   6 |  35 |   5 |  17 |   0 | 118 |
| WAYNESBORO            |  10 |   0 |   8 |   1 |   2 |   0 |   0 |   0 |   0 |   0 |   0 |   6 |  28 |   3 |   5 |   0 |  63 |
| WESTMORELAND          |  18 |   3 |   7 |   0 |   1 |   0 |   0 |   0 |   0 |   0 |   1 |   0 |  15 |   1 |   9 |   0 |  55 |
| WILLIAMSBURG CITY     |  12 |   0 |  13 |   0 |   4 |   0 |   0 |   0 |   0 |   3 |   2 |   2 |  78 |   1 |   6 |   0 | 121 |
| WINCHESTER CITY       |  12 |   0 |  14 |   1 |   9 |   0 |   1 |   0 |   0 |   0 |   0 |   3 |  46 |   0 |  10 |   0 |  96 |
| WISE                  |  22 |   1 |  18 |   1 |   5 |   0 |   0 |   0 |   0 |   0 |   0 |   2 |  28 |   2 |   2 |   0 |  81 |
| WYTHE                 |  19 |   0 |  13 |   0 |   2 |   0 |   0 |   0 |   0 |   1 |   0 |   9 |  24 |   8 |   5 |   0 |  81 |
| YORK                  |  27 |   2 |  29 |   0 |   5 |   0 |   0 |   0 |   0 |   1 |   0 |   8 |  97 |   8 |  28 |   0 | 205 |



labeled_formulas: J=G*H
J=G*I
J=G*L
J=H*I
J=(B*H)*I
J=(C*E)*G
J=(C*E)*H
J=(C*G)*H
J=(C*G)*I
J=(C*G)*K
J=(C*G)*L
J=(C*G)*O
J=(C*G)*Q
J=(C*H)*I
J=(C*I)*L
output: {"Arithmetic-Relationship": []}
source: $MMTU_HOME/resources/Relationship-finding/AR/table_298376___xlsx_f4cd3abbcafa6b7c2c4a81ca7c15f5508649f171.xlsx___line_872