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 |
|----:|:---------------|----------:|----------:|----------:|---------:|----------:|---------:|---------:|---------:|---------:|-----------------:|-----------------:|----:|
|   1 | Accomack       |    691719 |   8063571 |   4794131 |   153806 |  13549421 |  4183675 |  2860652 |     3027 |  4944711 |      1.24128e+06 |      1.02817e+06 |   1 |
|   2 | Albemarle      |    790881 |  17172200 |  29683519 |  1113269 |  47646600 | 16761562 |  8085377 |   276524 |  7825426 |      1.22864e+06 | 656201           |   2 |
|   3 | Alleghany      |    158325 |   6020823 |   4160793 |   144443 |  10339941 |  3841482 |  1821101 |        0 |  3846563 | 396940           | 439082           |   3 |
|   4 | Amelia         |    142055 |   3035277 |   1879832 |   153362 |   5057164 |  1332838 |   934553 |        0 |  2183574 | 364472           | 205669           |   4 |
|   5 | Amherst        |    239493 |   5817829 |   5848722 |   235745 |  11906044 |  3611108 |  2183386 |        0 |  4348269 | 471751           | 619399           |   5 |
|   6 | Appomattox     |    120000 |   1711057 |   4026623 |    51483 |   5857680 |  2259564 |  1248477 |        0 |  1280632 | 337502           | 416617           |   6 |
|   7 | Arlington      |  27850140 |  34135709 | 127603527 |  4264478 | 189589376 | 20310545 |  2325436 | 19019974 |  5854608 |      3.27713e+06 | 641425           |   7 |
|   8 | Augusta        |    571337 |   7947574 |  16538925 |   369623 |  25057836 |  8611570 |  7309372 |        0 |  4827537 |      1.15567e+06 | 927428           |   8 |
|   9 | Bath           |     99227 |    907504 |    939827 |    76943 |   1946558 |   577240 |   470381 |        0 |   483832 | 170739           |  62797.5         |   9 |
|  10 | Bedford        |         0 |         0 |         0 |        0 |         0 |        0 |        0 |        0 |        0 |      0           |      0           |  10 |
|  11 | Bland          |     88970 |   3817043 |   1391183 |    35533 |   5297196 |  1461029 |   780827 |    12940 |  3053542 | 306178           |  90787.5         |  11 |
|  12 | Botetourt      |    335098 |   1153796 |   4047746 |   523407 |   5536640 |  2115727 |  1116940 |    65374 |   472493 | 494874           | 188832           |  12 |
|  13 | Brunswick      |    114978 |   2577940 |   3085522 |    24455 |   5778440 |  2033109 |  1417121 |        0 |  1555311 | 324713           | 718383           |  13 |
|  14 | Buchanan       |    394791 |   3574348 |  10221377 |    48727 |  14190516 |  3969533 |  3943617 |  1610002 |  2522811 | 564358           |      1.81527e+06 |  14 |
|  15 | Buckingham     |    127935 |   1760296 |   3883083 |    28083 |   5771314 |  2389505 |  1259838 |        0 |  1266355 | 326705           | 461908           |  15 |
|  16 | Campbell       |    409181 |   7627431 |  12623042 |   384266 |  20659654 |  6994086 |  4601961 |        0 |  5700774 | 845814           |      1.21689e+06 |  16 |
|  17 | Caroline       |    275404 |   3612447 |   4860596 |   344643 |   8748447 |  2395616 |  1985453 |        0 |  2292624 | 617545           | 403604           |  17 |
|  18 | Carroll        |    262860 |  13383359 |   8565121 |   186400 |  22211340 |  7199074 |  3211116 |    45371 | 10706362 | 681822           | 906159           |  18 |
|  19 | Charles City   |    115869 |    123340 |   1704791 |   153340 |   1944000 |   606775 |   543558 |        0 |        0 | 327662           | 104229           |  19 |
|  20 | Charlotte      |    125573 |   3211306 |   4910925 |    22020 |   8247804 |  2176358 |  2606122 |        0 |  2310210 | 370242           | 437659           |  20 |
|  21 | Chesterfield   |   2623294 |  43646848 |  47608271 |  9169337 |  93878413 | 20918263 | 12186634 |   192753 | 24645756 |      3.48392e+06 |      1.79019e+06 |  21 |
|  22 | Clarke         |    199123 |   1414081 |   1814803 |   211007 |   3428007 |  1272126 |   765728 |    26522 |   695120 | 257305           |  50257.1         |  22 |
|  23 | Craig          |     45417 |    241441 |   1859074 |    38929 |   2145932 |   988059 |   617248 |     2612 |    98894 | 208049           | 100332           |  23 |
|  24 | Culpeper       |    402590 |   7520459 |  16379502 |   507590 |  24302551 |  6045996 |  6623138 |        0 |  4978555 | 705319           | 410874           |  24 |
|  25 | Cumberland     |     82233 |   1627040 |   2606126 |    55717 |   4315399 |  1461102 |   861455 |        0 |  1170490 | 358255           | 329951           |  25 |
|  26 | Dickenson      |    255524 |    204000 |   6973790 |   210794 |   7433314 |  2560306 |  2759072 |        0 |        0 | 415079           | 848548           |  26 |
|  27 | Dinwiddie      |    303275 |   1562463 |   5511050 |   612720 |   7376788 |  2802190 |  1640749 |        0 |   499004 | 505227           | 501490           |  27 |
|  28 | Essex          |    133000 |   1792567 |   3073632 |    91482 |   4999199 |  1669980 |  1460585 |        0 |   948556 | 340850           | 338696           |  28 |
|  29 | Fairfax        | 100337616 | 191156571 | 383307546 | 29000915 | 674801733 | 57260065 | 67974645 | 10631495 | 63050043 |      1.13425e+07 |      1.79674e+06 |  29 |
|  30 | Fauquier       |    462746 |   9866790 |  15388905 |  3367754 |  25718441 |  6786952 |  4872177 |        0 |  3197162 | 768220           | 267677           |  30 |
|  31 | Floyd          |    102189 |   3870746 |   3156974 |    43943 |   7129909 |  2654066 |  1198250 |   151129 |  2320863 | 276914           | 285819           |  31 |
|  32 | Fluvanna       |    278007 |   2715633 |   5973716 |   427989 |   8967356 |  3400132 |  1745510 |    43723 |  1217428 | 335162           | 216497           |  32 |
|  33 | Franklin       |    347156 |   4572023 |  15103566 |   307079 |  20022745 |  8025437 |  4642978 |    42052 |  2917433 | 722795           |      1.02381e+06 |  33 |
|  34 | Frederick      |    436439 |   6191264 |  12589755 |   808696 |  19217458 |  5796525 |  4618255 |   117567 |  2672455 | 752806           | 362983           |  34 |
|  35 | Giles          |    130254 |   6144572 |   6017909 |   100053 |  12292735 |  4408477 |  2165136 |   239905 |  3684167 | 297447           | 441182           |  35 |
|  36 | Gloucester     |    668478 |   6019570 |   5932913 |   292005 |  12620961 |  3729907 |  2802125 |        0 |  3185319 | 711906           | 437494           |  36 |
|  37 | Goochland      |    466641 |   2577044 |   4190973 |   480924 |   7234658 |  2243283 |  1772351 |        0 |   845681 | 365288           | 139576           |  37 |
|  38 | Grayson        |    178770 |   4226012 |   3574896 |    96816 |   7979678 |  2561476 |  1581029 |    14327 |  3380707 | 309393           | 568955           |  38 |
|  39 | Greene         |    257965 |   2231250 |   3222619 |   302549 |   5711834 |  2083005 |   918731 |    35924 |  1000278 | 299906           | 239054           |  39 |
|  40 | Greensville    |     85708 |   1100186 |   3023299 |    70075 |   4209193 |  1903740 |  1036482 |        0 |   351367 | 326952           | 259160           |  40 |
|  41 | Halifax        |    249199 |   6220991 |   9200411 |    31229 |  15670601 |  6361464 |  2969319 |        0 |  3753220 | 556617           |      1.32838e+06 |  41 |
|  42 | Hanover        |    653504 |  12044091 |  14304436 |  2047302 |  27002031 |  7686022 |  3623507 |        0 |  3029631 | 902089           | 485828           |  42 |
|  43 | Henrico        |   2537998 |  38499841 |  49978579 |  9942305 |  91016418 | 27185959 | 14710073 |        0 | 11795672 |      4.20722e+06 |      2.85265e+06 |  43 |
|  44 | Henry          |    315000 |  12355447 |   7152854 |   116305 |  19823301 |  6730996 |  3703093 |   113067 |  7799745 | 776095           |      1.5861e+06  |  44 |
|  45 | Highland       |     66319 |    384988 |    553353 |        0 |   1004660 |   251916 |   311307 |        0 |   224380 | 184245           |  29134.9         |  45 |
|  46 | Isle of Wight  |    890963 |  10611410 |   4194421 |   432041 |  15696794 |  5810303 |  4178047 |        0 |  6386529 | 572672           | 429522           |  46 |
|  47 | James City     |    738939 |   9766024 |   8717261 |  1805040 |  19222224 |  6241891 |  3323014 |        0 |  3089123 | 661403           | 394570           |  47 |
|  48 | King & Queen   |     61605 |   1296126 |   1737157 |    42830 |   3094888 |  1000817 |   808821 |        0 |   687950 | 196390           | 153812           |  48 |
|  49 | King George    |    308415 |   3633899 |   5298045 |   356160 |   9240359 |  2987410 |  1382835 |        0 |  2306238 | 660350           | 221446           |  49 |
|  50 | King William   |    183007 |   2624579 |   2432542 |   221647 |   5240128 |  1725465 |   920425 |        0 |  1412335 | 266228           | 170617           |  50 |
|  51 | Lancaster      |    312609 |   1834580 |   2690548 |    80614 |   4837737 |  1415113 |  1043834 |        0 |   973093 | 361547           | 277638           |  51 |
|  52 | Lee            |         0 |         0 |         0 |        0 |         0 |        0 |        0 |        0 |        0 |      0           |      0           |  52 |
|  53 | Loudoun        |   6831754 |  44717006 |  66178792 | 12002175 | 117727552 | 14833743 | 10748522 |  9730363 |  1614554 |      2.03087e+06 | 339327           |  53 |
|  54 | Louisa         |    688343 |   2844458 |   9881315 |   571133 |  13414116 |  4741161 |  2571567 |    45797 |  1275181 | 636198           | 503226           |  54 |
|  55 | Lunenburg      |    110000 |   2333158 |   2388136 |    25917 |   4831294 |  1637125 |   793272 |        0 |  1677032 | 392779           | 518214           |  55 |
|  56 | Madison        |    158700 |   1642686 |   4749301 |    65840 |   6550687 |  2697349 |  1553605 |        0 |   613487 | 297337           | 115190           |  56 |
|  57 | Mathews        |    138858 |   1448454 |   2154476 |   138322 |   3741788 |  1011028 |   956009 |        0 |   759586 | 268249           | 122843           |  57 |
|  58 | Mecklenburg    |    217509 |   5459586 |   6318351 |        0 |  11995446 |  4929285 |  2111145 |        0 |  3293852 | 518358           | 973081           |  58 |
|  59 | Middlesex      |    189990 |   1790233 |   2513068 |    92086 |   4493291 |  1528465 |   972216 |        0 |   947321 | 391115           | 263125           |  59 |
|  60 | Montgomery     |    596435 |  20679981 |   9299999 |   987759 |  30576415 |  8652600 |  4370960 |   807856 | 12419642 |      1.42087e+06 |      1.179e+06   |  60 |
|  61 | Nelson         |    248979 |   2117884 |   3566152 |   193791 |   5933015 |  2494508 |   913798 |    34099 |   949455 | 362164           | 328124           |  61 |
|  62 | New Kent       |    244312 |    135000 |   2712083 |   391807 |   3091395 |   642250 |   847475 |        0 |        0 | 484976           |  79823.5         |  62 |
|  63 | Northampton    |    417752 |   3066271 |   2514272 |    31519 |   5998295 |  1558234 |  1641360 |     1151 |  1880287 |      1.08245e+06 | 556724           |  63 |
|  64 | Northumberland |    177160 |   2192598 |   2105098 |    93003 |   4474856 |  1169026 |   959344 |    18187 |  1162228 | 416043           | 321517           |  64 |
|  65 | Nottoway       |    105000 |   1863843 |   2558245 |    29976 |   4527088 |  1460472 |   975896 |        0 |  1329081 | 385958           | 515099           |  65 |
|  66 | Orange         |    219636 |   4008741 |   7403200 |   189633 |  11631577 |  4123204 |  2323213 |        0 |  1870840 | 464355           | 359987           |  66 |
|  67 | Page           |    276082 |   1346110 |   4049578 |   241720 |   5671770 |  2016999 |  1445831 |    25561 |   610420 | 547344           | 451348           |  67 |
|  68 | Patrick        |    169083 |   4305930 |   2781796 |    54446 |   7256809 |  2357345 |  1424616 |    39099 |  2697154 | 427819           | 524349           |  68 |
|  69 | Pittsylvania   |    489264 |  10137484 |  12187320 |   283836 |  22814068 |  8808972 |  4954543 |    72715 |  5506127 |      1.16538e+06 |      1.40539e+06 |  69 |
|  70 | Powhatan       |    217638 |   2917923 |   3928152 |   486860 |   7063713 |  2618687 |  1303643 |        0 |   845682 | 370755           |  97670.2         |  70 |
|  71 | Prince Edward  |    181426 |   2638799 |   4003790 |    22170 |   6824015 |  1963196 |  1760294 |        0 |  1898349 | 513786           | 522491           |  71 |
|  72 | Prince George  |    222377 |   2144035 |   4918306 |   151637 |   7284718 |  2677827 |  1717405 |        0 |   684743 | 465943           | 258745           |  72 |
|  73 | Prince William |   3226000 |  40823000 |  82715000 | 22756000 | 126764000 | 35140000 | 19691000 |   415000 |   940000 |      3.61843e+06 |      1.28718e+06 |  73 |
|  74 | Pulaski        |    430849 |  15296866 |   9564192 |    84928 |  25291907 |  8558649 |  4137997 |   598811 |  9195809 | 681505           | 968776           |  74 |
|  75 | Rappahannock   |    138171 |    907561 |   3349484 |   129045 |   4395216 |  1383246 |  1030924 |        0 |   338943 | 221177           |  34553.4         |  75 |
|  76 | Richmond       |    139000 |   1448073 |   1384545 |    37729 |   2971618 |   916417 |   668968 |    15229 |   768231 | 342494           | 231759           |  76 |
|  77 | Roanoke        |    506261 |   7245537 |  20214053 |  1229937 |  27965851 | 10859036 |  6557531 |   207888 |  3074259 | 993191           | 810516           |  77 |
|  78 | Rockbridge     |    221514 |   4537519 |   5416257 |   223778 |  10175290 |  4703760 |  1237596 |        0 |  2419162 | 417948           | 510314           |  78 |
|  79 | Rockingham     |    692940 |   9187231 |  16930190 |   521034 |  26810361 |  9223093 |  5602246 |    72537 |  3106710 | 839062           | 632132           |  79 |
|  80 | Russell        |    340000 |   7186182 |   7114769 |   147137 |  14640951 |  4978577 |  3455484 |   144296 |  5043857 | 639414           |      1.38269e+06 |  80 |
|  81 | Scott          |    339842 |   3122404 |   4774975 |   187263 |   8237221 |  3193355 |  2632761 |        0 |  1040328 | 727212           | 848965           |  81 |
|  82 | Shenandoah     |    352599 |   3757047 |   9826230 |   286428 |  13935876 |  6034873 |  2241293 |    71324 |  1621295 | 700444           | 550740           |  82 |
|  83 | Smyth          |    441023 |  17028516 |   7068003 |   139968 |  24537542 |  6928994 |  3464289 |    57729 | 13622400 | 989311           |      1.24916e+06 |  83 |
|  84 | Southampton    |    316000 |   4234830 |   3347641 |   218085 |   7898471 |  2963208 |  1584330 |        0 |  2505133 | 588828           | 427016           |  84 |
|  85 | Spotsylvania   |    597513 |  13553113 |  24796985 |  1313709 |  38947611 | 11396227 |  7553857 |        0 |  8658017 |      1.46635e+06 | 947986           |  85 |
|  86 | Stafford       |    535937 |  12618613 |  18051234 |  1798587 |  31205784 |  7712079 |  5095669 |        0 |  8317514 |      1.42518e+06 | 662510           |  86 |
|  87 | Surry          |    209664 |   1280789 |   2834597 |    80377 |   4325050 |  1199165 |  1085422 |        0 |   409046 | 272491           | 116684           |  87 |
|  88 | Sussex         |    198317 |   1394511 |   2592791 |    35538 |   4185619 |  1496976 |  1251426 |        0 |   445029 | 422659           | 316583           |  88 |
|  89 | Tazewell       |    476773 |  14372364 |  10853839 |   268719 |  25702976 |  7708140 |  5084495 |   288591 | 10087714 | 932443           |      1.73477e+06 |  89 |
|  90 | Warren         |    352086 |   4766049 |   6094785 |   477464 |  11212920 |  3908332 |  2378233 |    89885 |  2043214 | 667071           | 408706           |  90 |
|  91 | Washington     |    501865 |  15804511 |   9130280 |   295966 |  25436656 |  7269918 |  4856040 |        0 |  9934965 |      1.06227e+06 |      1.11361e+06 |  91 |
|  92 | Westmoreland   |    180411 |   2891915 |   4252680 |    75971 |   7325006 |  2629121 |  1499868 |     4346 |  1530287 | 558157           | 508195           |  92 |
|  93 | Wise           |    514631 |   5163977 |  12112874 |   372724 |  17791482 |  7090134 |  6080045 |        0 |  1720542 |      1.19892e+06 |      1.78099e+06 |  93 |
|  94 | Wythe          |    380588 |  16856631 |   7434370 |   129724 |  24671589 |  7104561 |  3385190 |    57146 | 13484896 | 684630           | 695067           |  94 |
|  95 | York           |   1641589 |   4977270 |   7589270 |  1436658 |  14208129 |  3581827 |  3082528 |        0 |  1550719 | 540182           | 241815           |  95 |



label: G=(C+D+E)
labeled_formulas: G=(C+D+E)
output: {"Arithmetic-Relationship": ["G=(C+D+E)"]}
source: $MMTU_HOME/resources/Relationship-finding/AR/table_2319___xlsx_756f2de2f1b7907d0304e13012e31a03c6e2b186.xlsx___line_328