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:
| AP   | AQ   |     AR |   AS | AT   | AU   |     AV |   AW | AX   | AY   |     AZ |   BA | BB   | BC   |     BD |   BE |
|:-----|:-----|-------:|-----:|:-----|:-----|-------:|-----:|:-----|:-----|-------:|-----:|:-----|:-----|-------:|-----:|
| ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 |
| ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 |
| ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 |
| ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 |
| ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 |
| ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 |
| ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 |
| ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 |
| ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 |
| ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 |
| ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 |
| ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 |
| ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 |
| ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 |
| ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 |
| ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 |
| ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 |
| ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 |
| ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 |
| ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 |
| ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 |
| ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 |
| ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 |
| ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 |
| ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 |
| ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 |
| ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 |
| ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 |
| ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 |
| ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 |
| ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 |
| ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 |
| ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 |
| ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 |
| ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 |
| ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 |
| ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 |
| ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 |
| ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 |
| ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 |
| ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 |
| ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 |
| ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 |
| ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 | ---  | ---  | 999.99 |    0 |



labeled_formulas: AV=AR+AS
AZ=AR+AS
BD=AR+AS
AW=AR-AV
BA=AR-AV
BE=AR-AV
AZ=AR+AW
BD=AR+AW
BA=AR-AZ
BE=AR-AZ
BD=AR+BA
BE=AR-BD
AZ=AS+AV
BD=AS+AV
AZ=AV+AW
BD=AV+AW
BA=AV-AZ
BE=AV-AZ
BD=AV+BA
BE=AV-BD
AZ=(AR*AS)+AV
BD=(AR*AS)+AV
AW=(AR-AV)*AZ
AW=(AR-AV)*BD
BA=(AR-AV)*AZ
BD=(AR+AV)-AZ
BE=(AR-AV)*AZ
BA=(AR-AV)*BD
BE=(AR-AV)*BD
output: {"Arithmetic-Relationship": []}
source: $MMTU_HOME/resources/Relationship-finding/AR/table_108742___xlsx_73c87830003363b5a835392d0e1ae0dec3ce8d3f.xlsx___line_539