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 |
|----:|:---------------|----------:|---------:|---------:|---------:|---------:|-------:|----------:|----------:|---------:|---------:|--------:|-----:|----:|
|   3 | Alleghany      |  18312859 |  2562202 |  2291231 |  2935405 |  1083968 |   9230 |  27194895 |  14979789 |  2222024 |    42105 |  411246 |    0 |   3 |
|   4 | Amelia         |  12897697 |  1465137 |  1465306 |  1865319 |   715150 |   1095 |  18409704 |  11956577 |  1238315 |        0 |  193919 |    0 |   4 |
|   5 | Amherst        |  35664131 |  3061042 |  2931872 |  3645376 |  2109647 |   6742 |  47418810 |  29476764 |  4289734 |    63962 |  701478 |    0 |   5 |
|   9 | Bath           |   6811432 |   446746 |   894390 |  1272064 |   559929 |   5749 |   9990310 |   1813301 |   683684 |    29074 |   39772 |    0 |   9 |
|  10 | Bedford        |         0 |        0 |        0 |        0 |        0 |      0 |         0 |         0 |        0 |        0 |       0 |    0 |  10 |
|  11 | Bland          |   5483142 |   649092 |   515731 |  1183609 |   246035 |   9877 |   8087486 |   4758967 |   594571 |        0 |  134137 |    0 |  11 |
|  12 | Botetourt      |  40955024 |  2600292 |  3191215 |  5379974 |  1697651 |  42140 |  53866296 |  27016068 |  2549060 |        0 |  922796 |    0 |  12 |
|  13 | Brunswick      |  16070067 |   959283 |  1909365 |  2258673 |  1204469 |   9047 |  22410904 |  13031346 |  2987244 |    32568 |  112604 |    0 |  13 |
|  14 | Buchanan       |  22679293 |  2296111 |  2807715 |  4091781 |  2250268 | 300000 |  34425168 |  19095824 |  4775705 |        0 |  209730 |    0 |  14 |
|  15 | Buckingham     |  18323707 |  1183124 |  2274191 |  2270887 |  1065484 |  53290 |  25170683 |  14868720 |  3101830 |        0 |  126486 |    0 |  15 |
|  16 | Campbell       |  65240510 |  4406136 |  5069528 |  9181440 |  3599045 |      0 |  87496659 |  51211551 |  7104245 |        0 | 1726694 |    0 |  16 |
|  17 | Caroline       |  34098249 |  1653725 |  4046478 |  3928526 |  2086080 |   7237 |  45820295 |  27380992 |  3456361 |    63229 |  623110 |    0 |  17 |
|  19 | Charles City   |   6704242 |   977248 |  1078649 |  1634066 |   339699 |   1229 |  10735133 |   3703586 |   579155 |    76790 |   57134 |    0 |  19 |
|  20 | Charlotte      |  15235649 |  1250914 |  1425151 |  1950498 |   889941 |   5975 |  20758128 |  13631446 |  2837390 |        0 |  143599 |    0 |  20 |
|  22 | Clarke         |  18770751 |  1199617 |  1167505 |  2543033 |   854579 |  17796 |  24553281 |   9176805 |  1161384 |        0 |  590874 |    0 |  22 |
|  23 | Craig          |   5546463 |   578344 |   551334 |   625943 |   418090 |  17000 |   7737174 |   4493264 |   938470 |        0 |  125736 |    0 |  23 |
|  25 | Cumberland     |  11415485 |  1202281 |  1386412 |  1154431 |   921311 |   5593 |  16085513 |  10029749 |  1669967 |    26842 |   40642 |    0 |  25 |
|  26 | Dickenson      |  16353899 |  2186276 |  2229025 |  2853839 |  1708223 |  72567 |  25403829 |  15840295 |  3223504 |        0 |  111583 |    0 |  26 |
|  27 | Dinwiddie      |  36376759 |  2180148 |  2838316 |  5859280 |  1851705 |  10983 |  49117191 |  31165913 |  3350503 |        0 |  314655 |    0 |  27 |
|  28 | Essex          |  12767461 |  1100361 |  1583032 |  1599679 |   710022 |   8323 |  17768878 |   8488718 |  1721315 |        0 |   65793 |    0 |  28 |
|  30 | Fauquier       | 108547753 | 11986151 | 10513836 | 16837071 |  5533643 |  75197 | 153493651 |  48991954 |  5542041 |   850698 | 2361349 |    0 |  30 |
|  31 | Floyd          |  16429693 |   975925 |  1676937 |  2713966 |  1358901 |   7550 |  23162972 |  12896033 |  2082196 |        0 |  336476 |    0 |  31 |
|  33 | Franklin       |  61459295 |  3833737 |  6105778 |  7799255 |  3678043 |      0 |  82876108 |  42578112 |  7292210 |        0 | 2132542 |    0 |  33 |
|  34 | Frederick      | 143578296 |  7243830 | 10063987 | 15470048 |  5196020 |  81308 | 181633489 |  84098538 |  7867518 |        0 | 1949100 |    0 |  34 |
|  35 | Giles          |  19921723 |  1343211 |  1308473 |  2598357 |  1815852 |   9359 |  26996975 |  16987205 |  2400985 |        0 |  510649 |    0 |  35 |
|  37 | Goochland      |  26981836 |  2021600 |  2140971 |  2888566 |   993263 |      0 |  35026236 |   7780010 |  1721434 |        0 |  638540 | 8625 |  37 |
|  38 | Grayson        |  14289310 |  1155837 |  1936979 |  2211251 |  1047661 |  19412 |  20660450 |  11745575 |  2178066 |        0 |  163572 |    0 |  38 |
|  39 | Greene         |  27322015 |  1726060 |  2129210 |  2858792 |  1276203 |  40729 |  35353009 |  19839435 |  2253621 |        0 |  334041 |    0 |  39 |
|  41 | Halifax        |  39639307 |  3911819 |  4190873 |  4593557 |  2631355 |  97066 |  55063977 |  36690067 |  5700951 |    31622 |  439928 |    0 |  41 |
|  42 | Hanover        | 153172413 |  9569844 |  9367050 | 16756930 |  5175204 |      0 | 194041441 |  93182695 |  8094846 |  1319600 | 3714596 |    0 |  42 |
|  43 | Henrico        | 447906629 | 21144719 | 32146208 | 46760510 | 19122136 | 432386 | 567512588 | 298407378 | 36825169 |  1104237 | 5030669 |    0 |  43 |
|  44 | Henry          |  62005070 |  3340901 |  5345076 |  6256551 |  5362313 |  59442 |  82369353 |  56739883 | 10378535 |        0 |  883440 |    0 |  44 |
|  45 | Highland       |   3146375 |   227400 |   221858 |   409474 |   248593 |      0 |   4253700 |   1630808 |   500208 |        0 |   45276 |    0 |  45 |
|  46 | Isle of Wight  |  47453325 |  2378316 |  4043613 |  7973240 |  5925721 |      0 |  67774215 |  33143844 |  4873009 |        0 |  644469 |    0 |  46 |
|  47 | James City     | 106389141 |  8141317 |  6692826 | 11182756 |  3671737 |      0 | 136077777 |  47098857 |  7344897 |   150635 | 1664423 |    0 |  47 |
|  48 | King & Queen   |   7937554 |   907416 |  1077084 |   958177 |   323394 |   6294 |  11209919 |   5949677 |   801880 |    55879 |   73012 |    0 |  48 |
|  49 | King George    |  36890299 |  1461480 |  3426393 |  4187886 |  1811924 |   5774 |  47783756 |  27062006 |  2639682 |   269856 |  596551 |    0 |  49 |
|  50 | King William   |  18991742 |  1518422 |  2161634 |  2041302 |   861748 |   8635 |  25583483 |  12107145 |  1379629 |    55788 |  513772 |    0 |  50 |
|  51 | Lancaster      |  11901868 |  1055890 |  1381096 |  1795519 |   566913 |   7388 |  16708674 |   3612444 |  1162180 |    69507 |   61099 |    0 |  51 |
|  52 | Lee            |         0 |        0 |        0 |        0 |        0 |      0 |         0 |         0 |        0 |        0 |       0 |    0 |  52 |
|  54 | Louisa         |  47744913 |  3736805 |  4411585 |  6256526 |  2477646 |  56720 |  64684195 |  23664265 |  4494418 |        0 | 2158187 |    0 |  54 |
|  55 | Lunenburg      |  13289608 |   980011 |  1231205 |  1598121 |   815197 |      0 |  17914142 |  12598806 |  1839477 |        0 |   78716 |    0 |  55 |
|  56 | Madison        |  15541660 |  1165142 |  1508385 |  2166391 |  1009064 |   5000 |  21395642 |   9686356 |  1859048 |        0 |  261529 |    0 |  56 |
|  57 | Mathews        |  11083911 |  1084596 |  1277768 |  1530471 |   892919 |   7127 |  15876792 |   5789312 |   983485 |        0 |  325692 |    0 |  57 |
|  58 | Mecklenburg    |  37936292 |  2106486 |  5288881 |  3143705 |  2148592 |  18184 |  50642140 |  26491252 |  4357386 |        0 |  489745 |    0 |  58 |
|  59 | Middlesex      |  12384094 |  1116804 |  1307958 |  1121383 |   529695 |   6763 |  16466697 |   5330718 |  1093909 |        0 |  134472 |    0 |  59 |
|  60 | Montgomery     |  88472910 |  5116466 |  4603229 | 18118242 |  5126567 | 417341 | 121854755 |  60359371 |  8281934 |        0 | 1916263 |    0 |  60 |
|  61 | Nelson         |  18423864 |  1231250 |  2599428 |  2624583 |  1121708 |  11090 |  26011923 |   8969305 |  1609734 |   170539 |  329566 |    0 |  61 |
|  62 | New Kent       |  23569009 |  2513734 |  3048504 |  2606264 |  2590350 |   4999 |  34332860 |  18063365 |  1512827 |        0 |  678060 |    0 |  62 |
|  63 | Northampton    |  14371691 |  1549100 |  1185814 |  2065196 |   929428 |  20723 |  20121952 |   9349541 |  2215863 |   148081 |  245465 |    0 |  63 |
|  64 | Northumberland |  13768581 |   898506 |  1118297 |  2086558 |   687574 |   9104 |  18568620 |   4815525 |  1206783 |    51047 |   22051 |    0 |  64 |
|  66 | Orange         |  42735702 |  2197413 |  4557896 |  6536784 |  2240949 |  18600 |  58287344 |  29830997 |  4518197 |  1515331 |  575031 |    0 |  66 |
|  67 | Page           |  27451144 |  1853599 |  2021550 |  3153723 |  2038288 |  56217 |  36574521 |  22596412 |  3625074 |        0 | 1892516 |    0 |  67 |
|  68 | Patrick        |  20417623 |  1442968 |  1821697 |  2252075 |  1491418 |  17322 |  27443103 |  19229575 |  2516945 |        0 |  130393 |    0 |  68 |
|  69 | Pittsylvania   |  64131412 |  4050586 |  6668948 |  8346530 |  5337421 |  10428 |  88545325 |  62816989 |  9397311 |        0 | 1309419 |    0 |  69 |
|  70 | Powhatan       |  34724608 |  2184580 |  3917606 |  6099113 |  1165467 |      0 |  48091374 |  23362427 |  2027137 |        0 |  672288 |    0 |  70 |
|  71 | Prince Edward  |  19604804 |  1281081 |  1703472 |  1878503 |  1426811 |   7000 |  25901671 |  14731697 |  4200416 |        0 |  305509 |    0 |  71 |
|  72 | Prince George  |  53215650 |  3301787 |  4447535 |  6271171 |  3112786 |  16622 |  70365551 |  44560267 |  4123399 |  5608616 | 1053158 |    0 |  72 |
|  74 | Pulaski        |  35304509 |  1567102 |  2903815 |  5824202 |  2499170 | 107835 |  48206633 |  28563078 |  4857435 |        0 |  959796 |    0 |  74 |
|  75 | Rappahannock   |   9566722 |  1193404 |   997605 |  1152954 |   446907 |   6471 |  13364063 |   2882898 |   664529 |        0 |  132511 |    0 |  75 |
|  76 | Richmond       |  11773180 |   985303 |   839514 |  1469292 |   621622 |  10924 |  15699835 |   9443964 |  1403096 |    61813 |   32247 |    0 |  76 |
|  77 | Roanoke        | 118211875 |  5309468 |  7077360 | 14326102 | 14185603 |      0 | 159110408 |  83056623 |  8881107 |   265969 | 3197210 |    0 |  77 |
|  79 | Rockingham     | 115991948 |  6572595 |  9608138 | 13987733 |  5206361 |      0 | 151366775 |  70601835 |  9098453 |        0 | 5624700 | 5489 |  79 |
|  80 | Russell        |  30439725 |  2072416 |  2590440 |  4832326 |  2265920 | 365390 |  42566217 |  29745780 |  5778660 |        0 |  385747 |    0 |  80 |
|  81 | Scott          |  31891787 |  1890627 |  1643468 |  8181390 |  6907753 |  28019 |  50543044 |  29998216 |  4278833 |  2010360 | 4260721 |    0 |  81 |
|  83 | Smyth          |  35367548 |  1472385 |  2459468 |  5652703 |  2742770 |  56388 |  47751262 |  32556281 |  6076673 |        0 |  779572 |    0 |  83 |
|  84 | Southampton    |  23931850 |  1661586 |  3279265 |  3302348 |  1470730 |      0 |  33645779 |  19701580 |  2604763 |        0 |  403719 |    0 |  84 |
|  85 | Spotsylvania   | 211435659 | 11651865 | 19395422 | 22360975 | 10228619 | 189171 | 275261711 | 150431255 | 15673200 |  1308659 | 5790933 |    0 |  85 |
|  87 | Surry          |  11224703 |   943802 |  1236101 |  1966227 |   425233 |   1313 |  15797379 |   2421548 |   882543 |    69622 |  104266 |    0 |  87 |
|  88 | Sussex         |  12948785 |  1454690 |  1736091 |  1898615 |   843430 |   2995 |  18884606 |   8895796 |  1487195 |    82692 |  112591 |    0 |  88 |
|  89 | Tazewell       |  40955556 |  3420640 |  3241294 |  5479820 |  3118785 |  60000 |  56276095 |  40212415 |  6474212 |        0 |  931696 |    0 |  89 |
|  90 | Warren         |  46050125 |  2266695 |  2412357 |  6199278 |  2656541 |  41583 |  59626579 |  30532483 |  4606358 |        0 | 1419098 |    0 |  90 |
|  91 | Washington     |  61278764 |  3253086 |  4855444 |  7045082 |  3691753 | 217414 |  80341543 |  44293776 |  7247765 |        0 | 2207258 |    0 |  91 |
|  93 | Wise           |  42941516 |  2715768 |  3378936 |  6739019 |  3046081 |  54556 |  58875876 |  40419548 |  6740670 |        0 |  516013 |    0 |  93 |
|  94 | Wythe          |  32990315 |  1274746 |  2393833 |  4430746 |  2057445 |  22820 |  43169905 |  26334112 |  3817117 |        0 | 1089599 |    0 |  94 |
|  95 | York           | 116844342 |  9151648 |  8324800 | 14110046 |  3475360 |      0 | 151906196 |  72789874 |  5446658 | 13615584 | 2543093 |    0 |  95 |



labeled_formulas: O=(H*N)+A
O=(L*N)+A
output: {"Arithmetic-Relationship": []}
source: $MMTU_HOME/resources/Relationship-finding/AR/table_2322___xlsx_756f2de2f1b7907d0304e13012e31a03c6e2b186.xlsx___line_331