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:
|   CA |   CB | CC                          |               CD |      CE |       CF |       CG |       CH |   CI |       CJ |         CK |
|-----:|-----:|:----------------------------|-----------------:|--------:|---------:|---------:|---------:|-----:|---------:|-----------:|
|  189 |  189 | MILTON                      | 120491           |   61673 |  58818   |      0   |  21522   |    0 |  80340   |  56536.9   |
|  190 |  190 | MONROE                      |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  192 |  192 | MONTAGUE                    |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  193 |  193 | MONTEREY                    |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  194 |  194 | MONTGOMERY                  |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  195 |  195 | MOUNT WASHINGTON            |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  197 |  197 | NANTUCKET                   |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  200 |  200 | NEW ASHFORD                 |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  202 |  202 | NEW BRAINTREE               |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  203 |  205 | NEWBURY                     |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  205 |  203 | NEW MARLBOROUGH             |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  206 |  204 | NEW SALEM                   |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  207 |  207 | NEWTON                      | 136478           |   63439 |  73039   |      0   |  22762.4 |    0 |  95801.4 |  70206.4   |
|  216 |  217 | NORTHFIELD                  |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  219 |  219 | NORWELL                     | 272304           |  244060 |  28244   |  69560.4 |      0   |    0 |  97804.4 |  27148.6   |
|  222 |  222 | OAKHAM                      |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  223 |  223 | ORANGE                      |  39156           |   38340 |    816   |      0   |   8608   |    0 |   9424   |    784.354 |
|  224 |  224 | ORLEANS                     |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  225 |  225 | OTIS                        |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  228 |  228 | PAXTON                      |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  229 |  229 | PEABODY                     |      1.00306e+06 |  822963 | 180095   |  25662   |      0   |    0 | 205757   | 173111     |
|  230 |  230 | PELHAM                      |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  232 |  232 | PEPPERELL                   |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  233 |  233 | PERU                        |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  234 |  234 | PETERSHAM                   |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  235 |  235 | PHILLIPSTON                 |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  237 |  237 | PLAINFIELD                  |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  240 |  240 | PLYMPTON                    |      0           |       0 |      0   |      0   |   6234.8 |    0 |   6234.8 |      0     |
|  241 |  241 | PRINCETON                   |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  242 |  242 | PROVINCETOWN                | 311048           |  207793 | 103255   |      0   |  86627.2 |    0 | 189882   |  99250.6   |
|  243 |  243 | QUINCY                      | 920752           |  775025 | 145727   |   5037   |      0   |    0 | 150764   | 140075     |
|  244 |  244 | RANDOLPH                    |      5.21315e+06 | 5116415 |  96731.4 |      0   | 420372   |    0 | 517103   |  92980     |
|  245 |  245 | RAYNHAM                     |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  246 |  246 | READING                     |  21501           |   19518 |   1983   |      0   |      0   |    0 |   1983   |   1906.1   |
|  247 |  247 | REHOBOTH                    |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  249 |  249 | RICHMOND                    |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  250 |  250 | ROCHESTER                   |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  252 |  252 | ROCKPORT                    |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  254 |  254 | ROWLEY                      |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  255 |  255 | ROYALSTON                   |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  256 |  256 | RUSSELL                     |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  257 |  257 | RUTLAND                     |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  259 |  259 | SALISBURY                   |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  260 |  260 | SANDISFIELD                 |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  263 |  263 | SAVOY                       |  37941           |   37062 |    879   |      0   |      0   |    0 |    879   |    844.911 |
|  264 |  264 | SCITUATE                    | 324720           |  251630 |  73090   |      0   |      0   |    0 |  73090   |  70255.4   |
|  267 |  267 | SHEFFIELD                   |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  268 |  268 | SHELBURNE                   |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  269 |  269 | SHERBORN                    |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  270 |  270 | SHIRLEY                     |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  271 |  271 | SHREWSBURY                  | 450322           |  432948 |  17374   |  22404.6 |      0   |    0 |  39778.6 |  16700.2   |
|  272 |  272 | SHUTESBURY                  |  54990           |   35440 |  19550   |      0   |   9064   |    0 |  28614   |  18791.8   |
|  274 |  274 | SOMERVILLE                  |      7.98717e+06 | 7459791 | 527379   | 214966   |      0   |    0 | 742345   | 506926     |
|  279 |  279 | SOUTHWICK                   |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  280 |  280 | SPENCER                     |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  282 |  282 | STERLING                    |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  283 |  283 | STOCKBRIDGE                 |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  286 |  286 | STOW                        |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  287 |  287 | STURBRIDGE                  | 173952           |  136200 |  37752   |      0   |      0   |    0 |  37752   |  36287.9   |
|  288 |  288 | SUDBURY                     |  89728           |   60331 |  29397   |   4888.8 |      0   |    0 |  34285.8 |  28256.9   |
|  290 |  290 | SUTTON                      |  13119           |   12406 |    713   |   7443.6 |      0   |    0 |   8156.6 |    685.349 |
|  291 |  291 | SWAMPSCOTT                  | 777892           |  634311 | 143581   | 163453   |      0   |    0 | 307034   | 138013     |
|  294 |  294 | TEMPLETON                   |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  295 |  295 | TEWKSBURY                   | 996794           |  968975 |  27819   |      0   |  20053.6 |    0 |  47872.6 |  26740.1   |
|  297 |  297 | TOLLAND                     |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  298 |  298 | TOPSFIELD                   |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  299 |  299 | TOWNSEND                    |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  300 |  300 | TRURO                       | 132143           |  116884 |  15259   |      0   |   6056.8 |    0 |  21315.8 |  14667.2   |
|  301 |  301 | TYNGSBOROUGH                |      1.34172e+06 | 1082446 | 259279   |      0   |  49833.2 |    0 | 309112   | 249224     |
|  302 |  302 | TYRINGHAM                   |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  303 |  303 | UPTON                       |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  311 |  311 | WARREN                      |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  312 |  312 | WARWICK                     |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  313 |  313 | WASHINGTON                  |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  315 |  315 | WAYLAND                     |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  317 |  317 | WELLESLEY                   |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  318 |  318 | WELLFLEET                   |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  319 |  319 | WENDELL                     |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  320 |  320 | WENHAM                      |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  324 |  323 | WEST BROOKFIELD             |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  328 |  332 | WESTMINSTER                 |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  329 |  324 | WEST NEWBURY                |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  330 |  333 | WESTON                      |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  333 |  326 | WEST STOCKBRIDGE            |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  334 |  327 | WEST TISBURY                |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  338 |  338 | WHITMAN                     |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  339 |  339 | WILBRAHAM                   |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  341 |  341 | WILLIAMSTOWN                |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  343 |  343 | WINCHENDON                  | 253759           |  227561 |  26198   |      0   |      0   |    0 |  26198   |  25182     |
|  344 |  344 | WINCHESTER                  |  42864           |   18900 |  23964   |   4309.2 |      0   |    0 |  28273.2 |  23034.6   |
|  345 |  345 | WINDSOR                     |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  349 |  349 | WORTHINGTON                 |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  351 |  351 | YARMOUTH                    |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  406 |  406 | NORTHAMPTON SMITH           |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  616 |  616 | AYER SHIRLEY                | 949669           |  821426 | 128243   |      0   |      0   |    0 | 128243   | 123269     |
|  632 |  632 | CHESTERFIELD GOSHEN         |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  655 |  716 | DOVER SHERBORN              |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  662 |  788 | FARMINGTON RIVER            |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  685 |  727 | HAWLEMONT                   |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  698 |  698 | MANCHESTER ESSEX            |      0           |       0 |      0   |      0   |   6666   |    0 |   6666   |      0     |
|  728 |  787 | NEW SALEM WENDELL           |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  765 |  755 | SOUTHERN BERKSHIRE          |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  766 |  766 | SOUTHWICK TOLLAND GRANVILLE | 114651           |   67000 |  47651   |  10559.4 |      0   |    0 |  58210.4 |  45803     |
|  801 |  770 | ASSABET VALLEY              |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  805 |  708 | BLACKSTONE VALLEY           |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  806 |  709 | BLUE HILLS                  |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  810 |  771 | BRISTOL PLYMOUTH            |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  815 |  779 | CAPE COD                    |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  817 |  783 | ESSEX NORTH SHORE           |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  818 |  782 | FRANKLIN COUNTY             |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  821 |  722 | GREATER FALL RIVER          |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  823 |  723 | GREATER LAWRENCE            |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  825 |  786 | GREATER NEW BEDFORD         |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  828 |  767 | GREATER LOWELL              |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  829 |  778 | SOUTH MIDDLESEX             |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  830 |  781 | MINUTEMAN                   |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  832 |  735 | MONTACHUSETT                |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  851 |  743 | NORTHERN BERKSHIRE          |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  852 |  739 | NASHOBA VALLEY              |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  853 |  742 | NORTHEAST METROPOLITAN      |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  855 |  784 | OLD COLONY                  |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  860 |  773 | PATHFINDER                  |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  871 |  751 | SHAWSHEEN VALLEY            |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  872 |  754 | SOUTHEASTERN                |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  873 |  753 | SOUTH SHORE                 |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  876 |  762 | SOUTHERN WORCESTER          |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  878 |  785 | TRI COUNTY                  |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  879 |  758 | UPPER CAPE COD              |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  885 |  774 | WHITTIER                    |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  910 |  810 | BRISTOL COUNTY              |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |
|  915 |  830 | NORFOLK COUNTY              |      0           |       0 |      0   |      0   |      0   |    0 |      0   |      0     |



labeled_formulas: CF=CD-CE
CI=(CA-CB)*CF
CI=(CA-CB)*CG
CI=(CA-CB)*CH
CI=(CA-CB)*CK
CF=(CD-CE)+CI
CI=(CF*CG)*CH
CI=(CG*CH)*CK
output: {"Arithmetic-Relationship": []}
source: $MMTU_HOME/resources/Relationship-finding/AR/table_683761___xlsx_7bcfd6da22c8caf60b9f51defe0e3b19ce8ee549.xlsx___line_449