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 |
|:-------------|-----:|----:|----:|-----:|----:|-----:|-----:|----:|:-------------|-----:|----:|----:|----:|-----:|----:|----:|----:|----:|
| Albany       | 5182 |   0 |   0 | 3286 |   0 | 1461 |    0 |   0 | Albany       | 5182 |   0 |   0 |   0 |  278 |   0 |  96 |   0 |  61 |
| Broome       | 2999 |   0 |   0 |  110 |   0 |  772 | 2010 |   0 | Broome       | 2999 |   0 |   0 |   0 |    0 |   0 | 107 |   0 |   0 |
| Cayuga       | 1670 |   0 |   0 |    0 |   0 |  992 |  551 |   0 | Cayuga       | 1670 |   0 |   0 |   0 |    0 |   0 | 127 |   0 |   0 |
| Chemung      | 1027 |   0 |   0 |    0 |   0 |  660 |  292 |   0 | Chemung      | 1027 |   0 |   0 |   0 |    0 |   0 |  75 |   0 |   0 |
| Chenango     | 1043 |   0 |   0 |   26 |   0 |  634 |  345 |   0 | Chenango     | 1043 |   0 |   0 |   0 |    0 |   0 |  38 |   0 |   0 |
| Clinton      | 1557 |   0 |   0 |    1 |   0 | 1147 |  363 |   0 | Clinton      | 1557 |   0 |   0 |   0 |    0 |   0 |  46 |   0 |   0 |
| Columbia     | 1373 |   0 |   0 |  660 |   0 |  633 |    0 |   0 | Columbia     | 1373 |   0 |   0 |   0 |   66 |   0 |  14 |   0 |   0 |
| Delaware     |  702 |   0 |   0 |   59 |   0 |  486 |  157 |   0 | Delaware     |  702 |   0 |   0 |   0 |    0 |   0 |   0 |   0 |   0 |
| Dutchess     | 5613 |   0 |   0 |  123 |   0 | 2942 |    0 |   0 | Dutchess     | 5613 |   0 |   0 |   0 | 2134 |   0 |   0 |   0 | 414 |
| Essex        |  845 |   0 |   0 |   30 |   0 |  541 |  256 |   0 | Essex        |  845 |   0 |   0 |   0 |    0 |   0 |  18 |   0 |   0 |
| Franklin     |  769 |   0 |   0 |    0 |   0 |  501 |  268 |   0 | Franklin     |  769 |   0 |   0 |   0 |    0 |   0 |   0 |   0 |   0 |
| Fulton       | 1439 |   0 |   0 |  232 |   0 | 1158 |   44 |   0 | Fulton       | 1439 |   0 |   0 |   0 |    0 |   0 |   5 |   0 |   0 |
| Greene       |  975 |   0 |   0 |  475 |   0 |  460 |    0 |   0 | Greene       |  975 |   0 |   0 |   0 |   40 |   0 |   0 |   0 |   0 |
| Hamilton     |   93 |   0 |   0 |    0 |   0 |   80 |   13 |   0 | Hamilton     |   93 |   0 |   0 |   0 |    0 |   0 |   0 |   0 |   0 |
| Herkimer     | 1724 |   0 |   0 |   22 |   0 | 1305 |  389 |   0 | Herkimer     | 1724 |   0 |   0 |   0 |    0 |   0 |   8 |   0 |   0 |
| Jefferson    | 1998 |   0 |   0 |    0 |   0 |  281 |  884 |   0 | Jefferson    | 1998 |   0 |   0 |   0 |  112 |   0 | 721 |   0 |   0 |
| Lewis        |  866 |   0 |   0 |    0 |   0 |  291 |  427 |   0 | Lewis        |  866 |   0 |   0 |   0 |   29 |   0 | 119 |   0 |   0 |
| Livingston   | 1182 |   0 |   0 |    0 |   0 |  270 |  772 |   0 | Livingston   | 1182 |   0 |   0 |   0 |  140 |   0 |   0 |   0 |   0 |
| Madison      | 1296 |   0 |   0 |    4 |   0 |  800 |  408 |   0 | Madison      | 1296 |   0 |   0 |   0 |    0 |   0 |  84 |   0 |   0 |
| Montgomery   | 1277 |   0 |   0 |  365 |   0 |  870 |   42 |   0 | Montgomery   | 1277 |   0 |   0 |   0 |    0 |   0 |   0 |   0 |   0 |
| Oneida       | 4863 |   0 |   0 |   27 |   0 | 3090 | 1490 |   0 | Oneida       | 4863 |   0 |   0 |   0 |   48 |   0 | 208 |   0 |   0 |
| Otsego       | 1153 |   0 |   0 |  120 |   0 |  626 |  407 |   0 | Otsego       | 1153 |   0 |   0 |   0 |    0 |   0 |   0 |   0 |   0 |
| Rensselaer   | 2909 |   0 |   0 | 1934 |   0 |  751 |    0 |   0 | Rensselaer   | 2909 |   0 |   0 |   0 |  148 |   0 |  35 |   0 |  41 |
| St. Lawrence | 1825 |   0 |   0 |    0 |   0 |  875 |  692 |   0 | St. Lawrence | 1825 |   0 |   0 |   0 |    0 |   0 | 258 |   0 |   0 |
| Saratoga     | 4564 |   0 |   0 | 2467 |   0 | 1813 |    0 |   0 | Saratoga     | 4564 |   0 |   0 |   0 |  284 |   0 |   0 |   0 |   0 |
| Schenectady  | 3474 |   0 |   0 | 1622 |   0 | 1635 |    0 |   0 | Schenectady  | 3474 |   0 |   0 |   0 |  195 |   0 |   0 |   0 |  22 |
| Schoharie    |  617 |   0 |   0 |  424 |   0 |  193 |    0 |   0 | Schoharie    |  617 |   0 |   0 |   0 |    0 |   0 |   0 |   0 |   0 |
| Schuyler     |  345 |   0 |   0 |    0 |   0 |  227 |  118 |   0 | Schuyler     |  345 |   0 |   0 |   0 |    0 |   0 |   0 |   0 |   0 |
| Seneca       |  594 |   0 |   0 |    0 |   0 |  116 |  443 |   0 | Seneca       |  594 |   0 |   0 |   0 |    0 |   0 |  35 |   0 |   0 |
| Steuben      | 1504 |   0 |   0 |    0 |   0 |  874 |  625 |   0 | Steuben      | 1504 |   0 |   0 |   0 |    0 |   0 |   0 |   0 |   5 |
| Sullivan     | 1668 |   0 |   0 |    0 |   0 |  663 |    0 |   0 | Sullivan     | 1668 |   0 |   0 |   0 | 1005 |   0 |   0 |   0 |   0 |
| Tioga        |  828 |   0 |   0 |   76 |   0 |  429 |  291 |   0 | Tioga        |  828 |   0 |   0 |   0 |    0 |   0 |  32 |   0 |   0 |
| Ulster       | 4175 |   0 |   0 |  178 |   0 | 2328 |    0 |   0 | Ulster       | 4175 |   0 |   0 |   0 | 1395 |   0 |  91 |   0 | 183 |
| Warren       | 1616 |   0 |   0 |  316 |   0 | 1199 |    0 |   0 | Warren       | 1616 |   0 |   0 |   0 |   77 |   0 |  24 |   0 |   0 |
| Washington   | 1856 |   0 |   0 |  555 |   0 | 1253 |    0 |   0 | Washington   | 1856 |   0 |   0 |   0 |   48 |   0 |   0 |   0 |   0 |
| Wayne        | 2497 |   0 |   0 |    0 |   0 |  393 | 2034 |   0 | Wayne        | 2497 |   0 |   0 |   0 |    0 |   0 |  70 |   0 |   0 |
| Yates        |  497 |   0 |   0 |    0 |   0 |   80 |  417 |   0 | Yates        |  497 |   0 |   0 |   0 |    0 |   0 |   0 |   0 |   0 |



labeled_formulas: D=B*C
I=B*C
K=B+C
L=B*C
N=B*C
I=B*D
K=B+D
L=B*D
M=B*D
N=B*D
K=B+F
M=B*F
R=B*F
K=B+I
L=B*I
M=B*I
N=B*I
P=B*I
L=B-K
N=B-K
P=B-K
R=B-K
M=B*L
N=B*L
N=B*M
P=B*M
R=B*M
P=B*N
R=B*P
D=C*E
D=C*F
D=C*G
D=C*H
I=C*D
D=C*K
L=C*D
D=C*M
N=C*D
D=C*O
D=C*P
C=D*Q
D=C*R
C=D*S
F=C*E
I=C*E
L=C*E
M=C*E
N=C*E
P=C*E
R=C*E
D=(B*C)*E
D=(B*C)*F
D=(B*C)*G
D=(B*C)*H
I=(B*C)*D
K=(C*D)+B
L=(B*C)*D
D=(B*C)*M
N=(B*C)*D
D=(B*C)*O
D=(B*C)*P
C=(B*D)*Q
D=(B*C)*R
C=(B*D)*S
F=(B*C)*E
I=(B*C)*E
L=(B*C)*E
M=(B*C)*E
N=(B*C)*E
P=(B*C)*E
R=(B*C)*E
F=(B*C)*H
I=(B*C)*F
L=(B*C)*F
N=(B*C)*F
F=(B*C)*O
P=(B*C)*F
R=(B*C)*F
I=(B*C)*G
C=(B-K)*G
L=(B*C)*G
N=(B*C)*G
I=(B*C)*H
L=(B*C)*H
N=(B*C)*H
P=(B*C)*H
R=(B*C)*H
I=(B-K)*C
L=(B+C)*I
I=(B*C)*M
N=(B+C)*I
I=(B*C)*O
I=(B*C)*P
I=(B*C)*Q
I=(B*C)*R
I=(B*C)*S
L=(B-K)*C
N=(B-K)*C
K=(C*O)+B
C=(B-K)*Q
output: {"Arithmetic-Relationship": []}
source: $MMTU_HOME/resources/Relationship-finding/AR/table_214256___xlsx_66d155ef1c2aa83ca2625235b5f7d57d9123fe35.xlsx___line_313