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 |
|----:|:---------------|---------:|---------:|---------:|---------:|----:|---------:|-------:|--------:|--------:|---------:|---------:|--------:|-------:|--------:|----:|
|   4 | Amelia         |  2372790 |  2372790 |  1140836 |  1799524 |   0 |  1799524 |      0 |   97292 |  364311 |  5774753 |  1326637 |   39228 |      0 |  634901 |   4 |
|   5 | Amherst        |  3950956 |  3950956 |  2663475 |  3699804 |   0 |  3699804 |      0 |  284448 | 2390569 | 12989252 |  3609204 |  164219 |  53226 | 1633360 |   5 |
|   6 | Appomattox     |  1713768 |  1713768 |   291243 |  2028657 |   0 |  2028657 |      0 |   77498 |  940939 |  5052105 |  1880653 |   61548 |      0 |  234697 |   6 |
|   8 | Augusta        |  7502644 |  7502644 | 12677238 |  8627684 |   0 |  8627684 |      0 |  402287 | 2848044 | 32057897 |  6633295 |  153823 | 966733 | 2094802 |   8 |
|   9 | Bath           |  1074006 |  1074006 |   525410 |   174071 |   0 |   174071 |      0 |  194773 |  721327 |  2689587 |   563574 |     994 |  63959 |   21501 |   9 |
|  10 | Bedford        |        0 |        0 |        0 |        0 |   0 |        0 |      0 |       0 |       0 |        0 |        0 |       0 |      0 |       0 |  10 |
|  11 | Bland          |   829531 |   829531 |   240298 |   301445 |   0 |   301445 |      0 |   98613 |  316485 |  1786372 |   762882 |    7500 |      0 |   12558 |  11 |
|  14 | Buchanan       |  3051483 |  3051483 |   860364 |  4425724 |   0 |  4098703 |      0 |  119794 | 1604759 | 10062124 |  3106932 |  122522 |   3819 |  550219 |  14 |
|  15 | Buckingham     |  2018549 |  2018549 |  1228828 |  2238624 |   0 |  2238624 |      0 |  125903 |  197354 |  5809258 |  1594446 |   55194 |      0 |  791731 |  15 |
|  17 | Caroline       |  5702256 |  5702256 |  6475655 |  3584054 |   0 |  3584054 |      0 |  308716 |  880156 | 16950837 |  2825551 |  165641 | 924003 |  937312 |  17 |
|  18 | Carroll        |  3138069 |  3138069 |  2370462 |  4521962 |   0 |  4391490 |      0 |  280040 |  731260 | 11041793 |  4464549 |  126876 |      0 | 1620572 |  18 |
|  19 | Charles City   |  1203979 |  1203979 |  1630144 |   477421 |   0 |   477421 |      0 |  155131 |  240377 |  3707052 |   892874 |    8889 |      0 |  192214 |  19 |
|  22 | Clarke         |  3133626 |  3133626 |  1607883 |   863940 |   0 |   863940 |      0 |  205064 |  112611 |  5923124 |   938688 |   91014 | 115051 |  437120 |  22 |
|  23 | Craig          |   719803 |   719803 |   683086 |   146137 |   0 |   146137 |      0 |   83792 |   38796 |  1671614 |   441516 |   34868 |   2408 |  244818 |  23 |
|  25 | Cumberland     |  1494125 |  1494125 |   716464 |  1159730 |   0 |   849405 |      0 |  140482 |  306497 |  3817298 |  1099133 |     904 |      0 |  615192 |  25 |
|  26 | Dickenson      |  1965550 |  1965550 |   523886 |  2044698 |   0 |  2044698 |      0 |   94870 | 1152471 |  5781475 |  2094415 |  254083 |      0 |  185263 |  26 |
|  31 | Floyd          |  1845713 |  1845713 |  1794628 |  1022548 |   0 |  1022548 |      0 |  141959 |  193033 |  4997881 |  1617061 |   47252 |      0 |  626132 |  31 |
|  32 | Fluvanna       |  1492963 |  1492963 |  1055220 |  2372330 |   0 |   976814 |      0 |  287117 | 3688792 |  8896422 |  1433926 |   65550 |      0 | 1138892 |  32 |
|  33 | Franklin       |  4979140 |  4979140 |  4736339 |  7697169 |   0 |  7697169 |      0 |  443724 | 3089325 | 20945697 |  5286952 | 2577402 |      0 | 4682200 |  33 |
|  34 | Frederick      | 15617662 | 15617662 | 17694510 |  9820818 |   0 |  9820818 |      0 | 1396495 | 1512923 | 46042408 |  6978900 |  218546 | 142574 | 2954210 |  34 |
|  35 | Giles          |  1900825 |  1900825 |   873103 |  2244580 |   0 |  2074453 |      0 |  130073 |  434254 |  5582835 |  2536744 |   64793 |      0 |   60340 |  35 |
|  37 | Goochland      |  3657317 |  3657317 |  7721366 |   592171 |   0 |   592171 |      0 |  767719 | 1956965 | 14695538 |  1130104 |   38643 |      0 |  754581 |  37 |
|  38 | Grayson        |  1632067 |  1632067 |   690253 |  1419477 |   0 |  1419477 |      0 |  164690 |  139071 |  4045558 |  2005122 |  238344 |      0 |    8335 |  38 |
|  39 | Greene         |  3297143 |  3297143 |  1715214 |  2555058 |   0 |  2372624 |      0 |  319179 |  414564 |  8301158 |  1611269 |   68414 |      0 |  893088 |  39 |
|  40 | Greensville    |  2139545 |  2139545 |   315830 |  2507703 |   0 |  2469458 |  38245 |  170358 |  198782 |  5332218 |  1993986 |  150494 |      0 |  212099 |  40 |
|  41 | Halifax        |  6327633 |  6327633 |  1141270 |  6045253 |   0 |  5517771 |      0 |  220274 |  299399 | 14033829 |  4777902 |   17963 |  80343 |  586515 |  41 |
|  45 | Highland       |   746193 |   746193 |   268208 |    64286 |   0 |    64286 |      0 |   86235 |  229989 |  1394911 |   538628 |    7829 |  21176 |   21738 |  45 |
|  46 | Isle of Wight  |  3712794 |  3712794 |  5692703 |  1784390 |   0 |  1784390 |      0 |  453480 | 4141221 | 15784588 |  2529235 |  284945 |      0 | 1746893 |  46 |
|  48 | King & Queen   |  1543703 |  1543703 |  1463854 |   968747 |   0 |   968747 |      0 |   92089 |  220525 |  4288918 |   669397 |   28383 |      0 |  154274 |  48 |
|  49 | King George    |  3595870 |  3595870 |  4977345 |  2100482 |   0 |  1110947 |      0 |  553038 | 1225966 | 12452701 |  1856747 |  111188 | 279236 |  319615 |  49 |
|  50 | King William   |  2036605 |  2036605 |  2314308 |  1063110 |   0 |  1063110 |      0 |  186363 |  359009 |  5959395 |   797668 |   51649 |      0 |  163132 |  50 |
|  52 | Lee            |        0 |        0 |        0 |        0 |   0 |        0 |      0 |       0 |       0 |        0 |        0 |       0 |      0 |       0 |  52 |
|  54 | Louisa         |  5338614 |  5338614 |  7125066 |  5216538 |   0 |  2170846 |      0 |  674623 | 1199789 | 19554630 |  2930417 |  117798 |  65369 | 2245117 |  54 |
|  55 | Lunenburg      |  1069489 |  1069489 |   315552 |  1791070 |   0 |  1791070 |      0 |   91449 |  181438 |  3448998 |  1271727 |  162042 |      0 |  638143 |  55 |
|  57 | Mathews        |  1441651 |  1441651 |   550645 |   625604 |   0 |   625604 |      0 |  149602 |  103294 |  2870796 |   659640 |   44366 |      0 |   27214 |  57 |
|  58 | Mecklenburg    |  9334973 |  9334973 |  1477740 |  3669097 |   0 |  3669097 |      0 |  252004 | 1970363 | 16704177 |  5423492 |   98263 |  14626 |  881673 |  58 |
|  59 | Middlesex      |  1789176 |  1789176 |   378528 |  1232636 |   0 |  1232636 |      0 |   87019 |  788927 |  4276286 |  1034014 |   12901 |      0 |   20364 |  59 |
|  61 | Nelson         |  1941966 |  1941966 |  2422975 |  1246594 |   0 |  1246594 |      0 |  305005 |  270203 |  6186743 |  1271192 |   28435 |  22138 |   79287 |  61 |
|  62 | New Kent       |  3511680 |  3511680 |  4188959 |  1296644 |   0 |    88282 | 100847 |  479630 | 1748377 | 11225290 |   913622 |  427349 |      0 |    6323 |  62 |
|  64 | Northumberland |  1953674 |  1953674 |  2024428 |  1955767 |   0 |  1955767 |      0 |  230257 |  433806 |  6597932 |  1403345 |    3010 |      0 | 1435084 |  64 |
|  65 | Nottoway       |  1656832 |  1656832 |   511290 |  3214879 |   0 |  3214879 |      0 |   83736 |  145997 |  5612734 |  1770034 |   43137 |      0 | 1280655 |  65 |
|  66 | Orange         |  2307083 |  2307083 | 10142218 |  4179787 |   0 |  4176194 |      0 |  308693 | 4710719 | 21648500 |  2841664 |   74261 |      0 | 2141908 |  66 |
|  70 | Powhatan       |  4954141 |  4954141 |  2416856 |   812054 |   0 |   812054 |      0 |  361555 | 3553679 | 12098285 |  1434296 |  331854 |  19235 |  685507 |  70 |
|  71 | Prince Edward  |  2344727 |  2344727 |  1182691 |  4266007 |   0 |  3489555 |      0 |   64577 |  179217 |  8037219 |  2440185 |  101203 |      0 | 1505569 |  71 |
|  74 | Pulaski        |  3366851 |  3366851 |  2458685 |  4975175 |   0 |  4015726 |      0 |  243080 | 1596668 | 12640459 |  5763175 |  271672 |      0 | 1380243 |  74 |
|  75 | Rappahannock   |  1520319 |  1520319 |  1266664 |   284899 |   0 |   284899 |      0 |  207376 |  264602 |  3543860 |   966270 |    5844 |      0 |  114882 |  75 |
|  76 | Richmond       |  1252840 |  1252840 |   990094 |  1943968 |   0 |  1943968 |      0 |   82271 |  333130 |  4602303 |  1310023 |   80541 |      0 | 1397843 |  76 |
|  78 | Rockbridge     |  2646656 |  2646656 |  5102954 |  3961397 |   0 |  3726744 |      0 |  240322 |  418832 | 12370161 |  2810993 |  131895 | 161194 |  216871 |  78 |
|  81 | Scott          |  3123102 |  3123102 |   724686 |  4155502 |   0 |  4155502 |      0 |   90505 |  185791 |  8279586 |  3457084 |   19685 |   3661 |  548682 |  81 |
|  83 | Smyth          |  4275661 |  4275661 |   637710 |  5353677 |   0 |  5353677 |      0 |  275093 |  778685 | 11320826 |  4417485 |   91357 |      0 |  598011 |  83 |
|  85 | Spotsylvania   | 24029294 | 24029294 | 32679743 | 15372756 |   0 | 15372756 |      0 | 2776995 | 1641870 | 76500658 | 10043079 | 9884446 | 263822 | 5408516 |  85 |
|  86 | Stafford       | 26547520 | 26547520 | 23549170 | 16828971 |   0 | 16538750 | 290221 | 4747033 | 1183242 | 72855936 | 11783638 |  684519 |      0 | 4055130 |  86 |
|  87 | Surry          |  1712062 |  1712062 |   771089 |   523084 |   0 |   523084 |      0 |  131206 |  678397 |  3815838 |   683323 |   57992 |      0 |  241726 |  87 |
|  89 | Tazewell       |  5635802 |  5635802 |  2257850 |  7394035 |   0 |  7091805 | 302230 |  249733 |  926053 | 16463473 |  5321840 |  230761 |      0 |  615366 |  89 |
|  90 | Warren         |  5441197 |  5441197 |  4839263 |  7163131 |   0 |  7119796 |  43335 |  669824 |  713043 | 18826458 |  4702710 |   67967 |      0 | 2251820 |  90 |
|  91 | Washington     |  6169095 |  6169095 |  2392466 |  7070959 |   0 |  7070959 |      0 |  309564 |  891831 | 16833915 |  5235254 |  181385 |      0 |  586915 |  91 |
|  92 | Westmoreland   |  2851264 |  2851264 |  1876249 |  2278270 |   0 |  2045272 |      0 |  215129 |  593128 |  7814040 |  1981725 |  128671 |      0 | 1123839 |  92 |
|  93 | Wise           |  6255099 |  6255099 |   591055 |  8224098 |   0 |  7373155 |      0 |  153099 |  495916 | 15719267 |  6468335 |  240604 |   9243 | 1022919 |  93 |
|  94 | Wythe          |  2626747 |  2626747 |  1061422 |  3175127 |   0 |  3173448 |   1679 |  100940 |  882904 |  7847140 |  3814480 |  240486 |      0 |   10145 |  94 |
|  95 | York           | 11481813 | 11481813 | 14603112 |  6044155 |   0 |  3320789 |      0 | 1157848 | 7103115 | 40390043 |  5848830 |  762273 | 239504 | 2134235 |  95 |



labeled_formulas: D=(A/E)+C
D=(A/F)+C
D=(A/J)+C
D=(A/K)+C
D=(A/L)+C
D=(A/M)+C
Q=(A+C)-D
Q=(D*G)+A
Q=(G*H)+A
Q=(G*I)+A
Q=(G*O)+A
Q=(I*O)+A
D=(Q/E)+C
D=(Q/F)+C
D=(G*I)+C
D=(G*O)+C
D=(Q/J)+C
D=(Q/K)+C
D=(Q/L)+C
D=(Q/M)+C
output: {"Arithmetic-Relationship": []}
source: $MMTU_HOME/resources/Relationship-finding/AR/table_2313___xlsx_756f2de2f1b7907d0304e13012e31a03c6e2b186.xlsx___line_322