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 |
|----:|:---------------|---------:|---------:|----:|---------:|---------:|----:|---------:|---------:|---------:|----:|--------:|---------:|----:|----:|-------:|---------:|----:|
|   2 | Albemarle      |   415881 | 23796833 |   0 | 24212714 | 10185133 |   0 |  6409090 | 16594223 |  4857368 |   0 | 2794472 |  7651840 |   0 |   0 |      0 | 24246063 |   2 |
|   3 | Alleghany      |        0 |  1878480 |   0 |  1878480 |  1273339 |   0 |   221674 |  1495013 |   143460 |   0 |  240007 |   383467 |   0 |   0 |      0 |  1878480 |   3 |
|   4 | Amelia         |    60446 |        0 |   0 |    60446 |   477446 |   0 |        0 |   477446 |   183469 |   0 |       0 |   183469 |   0 |   0 |      0 |   660915 |   4 |
|   5 | Amherst        |        0 |        0 |   0 |        0 |  1838916 |   0 |   798097 |  2637013 |   661694 |   0 |  279947 |   941641 |   0 |   0 |      0 |  3578654 |   5 |
|   6 | Appomattox     |   434475 |        0 |   0 |   434475 |  1293541 |   0 |   993056 |  2286597 |   504224 |   0 |  324528 |   828752 |   0 |   0 |      0 |  3115349 |   6 |
|   9 | Bath           |        0 |  1254406 |   0 |  1254406 |   101187 |   0 |  1020000 |  1121187 |    55163 |   0 |   78056 |   133219 |   0 |   0 |      0 |  1254406 |   9 |
|  10 | Bedford        |        0 |        0 |   0 |        0 |        0 |   0 |        0 |        0 |        0 |   0 |       0 |        0 |   0 |   0 |      0 |        0 |  10 |
|  11 | Bland          |        0 |        0 |   0 |        0 |        0 |   0 |   145000 |   145000 |        0 |   0 |   77759 |    77759 |   0 |   0 |      0 |   222759 |  11 |
|  12 | Botetourt      |        0 |  4912092 |   0 |  4912092 |  1908622 |   0 |   931848 |  2840470 |  1509388 |   0 |  562234 |  2071622 |   0 |   0 |      0 |  4912092 |  12 |
|  13 | Brunswick      |    56938 |        0 |   0 |    56938 |   552086 |   0 |   557736 |  1109822 |   203904 |   0 |  413894 |   617798 |   0 |   0 |      0 |  1727620 |  13 |
|  14 | Buchanan       |        0 |  1575502 |   0 |  1575502 |  1297388 |   0 |    68697 |  1366085 |   198317 |   0 |   11100 |   209417 |   0 |   0 |      0 |  1575502 |  14 |
|  15 | Buckingham     |   489639 |        0 |   0 |   489639 |  2301810 |   0 |   105000 |  2406810 |  1325855 |   0 |   92789 |  1418644 |   0 |   0 | 104186 |  3929640 |  15 |
|  16 | Campbell       |   942248 |  1899027 |   0 |  2841275 |  1117234 |   0 |  2000000 |  3117234 |   238701 |   0 |  790850 |  1029551 |   0 |   0 |      0 |  4146785 |  16 |
|  17 | Caroline       |  4843975 |    41420 |   0 |  4885395 |  2647663 |   0 |  7107815 |  9755478 |  2051787 |   0 | 1069347 |  3121134 |   0 |   0 | 103152 | 12979764 |  17 |
|  18 | Carroll        |   651712 |  4375004 |   0 |  5026716 |  3496769 |   0 |   235691 |  3732460 |  1120217 |   0 |  174039 |  1294256 |   0 |   0 |      0 |  5026716 |  18 |
|  19 | Charles City   |        0 |        0 |   0 |        0 |        0 |   0 |   329858 |   329858 |        0 |   0 |   54715 |    54715 |   0 |   0 |      0 |   384573 |  19 |
|  20 | Charlotte      |        0 |        0 |   0 |        0 |   167320 |   0 |   466000 |   633320 |   202239 |   0 |  517000 |   719239 |   0 |   0 |      0 |  1352559 |  20 |
|  22 | Clarke         |   223149 |  2547656 |   0 |  2770805 |  1526751 |   0 |    92013 |  1618764 |   992354 |   0 |  159687 |  1152041 |   0 |   0 |      0 |  2770805 |  22 |
|  23 | Craig          |        0 |   728872 |   0 |   728872 |   312610 |   0 |   267255 |   579865 |    90390 |   0 |   58617 |   149007 |   0 |   0 |      0 |   728872 |  23 |
|  24 | Culpeper       |    42040 |   344705 |   0 |   386745 |  5287082 |   0 |  1380449 |  6667531 |  2835957 |   0 |  322485 |  3158442 |   0 |   0 |      0 |  9825973 |  24 |
|  25 | Cumberland     |  3060000 |        0 |   0 |  3060000 |  1513231 |   0 |  3548222 |  5061453 |   921339 |   0 |  191399 |  1112738 |   0 |   0 |   8520 |  6182711 |  25 |
|  27 | Dinwiddie      | 23490000 |  6490216 |   0 | 29980216 |  2631571 |   0 | 24859808 | 27491379 |  1310992 |   0 | 1155012 |  2466004 |   0 |   0 |      0 | 29957383 |  27 |
|  30 | Fauquier       |  9502622 | 11953413 |   0 | 21456035 | 16383804 |   0 |  1964594 | 18348398 |  2509757 |   0 | 1410380 |  3920137 |   0 |   0 |  52900 | 22321435 |  30 |
|  31 | Floyd          |        0 |  2334387 |   0 |  2334387 |  1011571 |   0 |   666055 |  1677626 |   210325 |   0 |  446436 |   656761 |   0 |   0 |      0 |  2334387 |  31 |
|  32 | Fluvanna       |   201087 |  8873953 |   0 |  9075040 |  4435154 |   0 |  1563885 |  5999039 |  2929055 |   0 |  146946 |  3076001 |   0 |   0 |      0 |  9075040 |  32 |
|  33 | Franklin       |        0 |  6160993 |   0 |  6160993 |   998211 |   0 |  3456750 |  4454961 |   193164 |   0 | 1370539 |  1563703 |   0 |   0 | 142329 |  6160993 |  33 |
|  34 | Frederick      |   584844 | 19301721 |   0 | 19886565 | 11195047 |   0 |  1672359 | 12867406 |  5388530 |   0 | 1330236 |  6718766 |   0 |   0 |      0 | 19586172 |  34 |
|  35 | Giles          |  7500000 |  1098835 |   0 |  8598835 |  7463321 |   0 |   569514 |  8032835 |   222725 |   0 |  212845 |   435570 |   0 |   0 | 130430 |  8598835 |  35 |
|  36 | Gloucester     |   222975 |        0 |   0 |   222975 |  2342866 |   0 |   580065 |  2922931 |  1095330 |   0 |  154046 |  1249376 |   0 |   0 |      0 |  4172307 |  36 |
|  37 | Goochland      |        0 |  3098603 |   0 |  3098603 |  2049789 |   0 |  1048814 |  3098603 |   192640 |   0 |  143970 |   336610 |   0 |   0 |      0 |  3435213 |  37 |
|  38 | Grayson        |        0 |  1454572 |   0 |  1454572 |   986046 |   0 |     4482 |   990528 |   462733 |   0 |    1311 |   464044 |   0 |   0 |      0 |  1454572 |  38 |
|  39 | Greene         | 14727907 |  1714977 |   0 | 16442884 |  1750893 |   0 | 13866867 | 15617760 |  1405318 |   0 | 1798717 |  3204035 |   0 |   0 |      0 | 18821795 |  39 |
|  40 | Greensville    |        0 |        0 |   0 |        0 |   799000 |   0 |   882095 |  1681095 |   263013 |   0 |  463707 |   726720 |   0 |   0 |      0 |  2407815 |  40 |
|  41 | Halifax        |        0 |        0 |   0 |        0 |  3769932 |   0 |   837473 |  4607405 |  1189091 |   0 | 1040423 |  2229514 |   0 |   0 |      0 |  6836919 |  41 |
|  42 | Hanover        | 18107831 | 19663830 |   0 | 37771661 |  9975281 |   0 | 21228114 | 31203395 |  2782491 |   0 | 3428812 |  6211303 |   0 |   0 | 328158 | 37742856 |  42 |
|  44 | Henry          |        0 |        0 |   0 |        0 |  2473315 |   0 |        0 |  2473315 |   917135 |   0 | 2592699 |  3509834 |   0 |   0 |      0 |  5983149 |  44 |
|  45 | Highland       |        0 |   114530 |   0 |   114530 |   110694 |   0 |        0 |   110694 |     3836 |   0 |       0 |     3836 |   0 |   0 |      0 |   114530 |  45 |
|  46 | Isle of Wight  |   529624 |        0 |   0 |   529624 |  3777038 |   0 |  3639410 |  7416448 |  1934335 |   0 | 2546151 |  4480486 |   0 |   0 |  15350 | 11912284 |  46 |
|  47 | James City     |   100343 | 19673916 |   0 | 19774259 | 10059260 |   0 |  4223581 | 14282841 |  4237108 |   0 | 1210171 |  5447279 |   0 |   0 |  44139 | 19774259 |  47 |
|  48 | King & Queen   |        0 |        0 |   0 |        0 |    71895 |   0 |        0 |    71895 |    37847 |   0 |       0 |    37847 |   0 |   0 |      0 |   109742 |  48 |
|  49 | King George    | 19060000 |  6440149 |   0 | 25500149 | 21469911 |   0 |   983518 | 22453429 |  2097858 |   0 |  748877 |  2846735 |   0 |   0 | 199985 | 25500149 |  49 |
|  50 | King William   |    51335 |   756774 |   0 |   808109 |   943400 |   0 |   568281 |  1511681 |   650305 |   0 |  258694 |   908999 |   0 |   0 |  41735 |  2462415 |  50 |
|  51 | Lancaster      |    15012 |        0 |   0 |    15012 |   160000 |   0 |   933736 |  1093736 |    60254 |   0 |  185025 |   245279 |   0 |   0 |      0 |  1339015 |  51 |
|  52 | Lee            |        0 |        0 |   0 |        0 |        0 |   0 |        0 |        0 |        0 |   0 |       0 |        0 |   0 |   0 |      0 |        0 |  52 |
|  54 | Louisa         |        0 |        0 |   0 |        0 |  1945442 |   0 |   695000 |  2640442 |   991666 |   0 | 1512909 |  2504575 |   0 |   0 |      0 |  5145017 |  54 |
|  55 | Lunenburg      |   143253 |  1108738 |   0 |  1251991 |   767579 |   0 |   457000 |  1224579 |   306922 |   0 |   56775 |   363697 |   0 |   0 |    625 |  1588901 |  55 |
|  57 | Mathews        |        0 |        0 |   0 |        0 |   491600 |   0 |   510000 |  1001600 |    10512 |   0 |   89624 |   100136 |   0 |   0 |      0 |  1101736 |  57 |
|  58 | Mecklenburg    |        0 |  6849514 |   0 |  6849514 |  4368631 |   0 |        0 |  4368631 |  2480883 |   0 |       0 |  2480883 |   0 |   0 |      0 |  6849514 |  58 |
|  59 | Middlesex      |        0 |        0 |   0 |        0 |  1038498 |   0 |  1099881 |  2138379 |   338359 |   0 |  242660 |   581019 |   0 |   0 |      0 |  2719398 |  59 |
|  60 | Montgomery     |        0 |        0 |   0 |        0 | 12392978 |   0 |  3718209 | 16111187 |  6158852 |   0 | 1031087 |  7189939 |   0 |   0 |  14423 | 23315549 |  60 |
|  61 | Nelson         |        0 |  3351467 |   0 |  3351467 |  1604718 |   0 |   890000 |  2494718 |   433546 |   0 |  422106 |   855652 |   0 |   0 |      0 |  3350370 |  61 |
|  62 | New Kent       |  3021000 |        0 |   0 |  3021000 |  3550474 |   0 |  4191818 |  7742292 |  1624059 |   0 |  313591 |  1937650 |   0 |   0 |  81596 |  9761538 |  62 |
|  63 | Northampton    |   237427 |  2829193 |   0 |  3066620 |   235792 |   0 |  2023684 |  2259476 |    39239 |   0 |  738497 |   777736 |   0 |   0 | 126568 |  3163780 |  63 |
|  64 | Northumberland |        0 |        0 |   0 |        0 |  1050000 |   0 |        0 |  1050000 |   977615 |   0 |       0 |   977615 |   0 |   0 |      0 |  2027615 |  64 |
|  65 | Nottoway       |        0 |        0 |   0 |        0 |   341252 |   0 |        0 |   341252 |    80618 |   0 |       0 |    80618 |   0 |   0 |      0 |   421870 |  65 |
|  67 | Page           |        0 |  5836752 |   0 |  5836752 |  2607975 |   0 |   840099 |  3448074 |  1889671 |   0 |  499007 |  2388678 |   0 |   0 |      0 |  5836752 |  67 |
|  68 | Patrick        |    80827 |  2440353 |   0 |  2521180 |   965289 |   0 |    70464 |  1035753 |  1055912 |   0 |  429515 |  1485427 |   0 |   0 |      0 |  2521180 |  68 |
|  69 | Pittsylvania   |    81014 |        0 |   0 |    81014 |  6422346 |   0 |  1563551 |  7985897 |  2362109 |   0 |  351329 |  2713438 |   0 |   0 |      0 | 10699335 |  69 |
|  71 | Prince Edward  |  7208469 |        0 |   0 |  7208469 |   364694 |   0 |   813215 |  1177909 |   128625 |   0 |  220978 |   349603 |   0 |   0 |      0 |  1527512 |  71 |
|  72 | Prince George  |        0 |        0 |   0 |        0 |  2227512 |   0 |  4271757 |  6499269 |   409559 |   0 | 1029006 |  1438565 |   0 |   0 | 134223 |  8072057 |  72 |
|  74 | Pulaski        |        0 |  6421927 |   0 |  6421927 |  3890630 |   0 |   300493 |  4191123 |  2157401 |   0 |   73403 |  2230804 |   0 |   0 |      0 |  6421927 |  74 |
|  75 | Rappahannock   |        0 |   384730 |   0 |   384730 |   320000 |   0 |        0 |   320000 |    64730 |   0 |       0 |    64730 |   0 |   0 |      0 |   384730 |  75 |
|  76 | Richmond       |   255850 |        0 |   0 |   255850 |   647213 |   0 |   488137 |  1135350 |   496029 |   0 |  150541 |   646570 |   0 |   0 |  63443 |  1845363 |  76 |
|  78 | Rockbridge     |        0 |  5077223 |   0 |  5077223 |  2523090 |   0 |   585000 |  3108090 |  1147337 |   0 |  821796 |  1969133 |   0 |   0 |      0 |  5077223 |  78 |
|  79 | Rockingham     | 31544588 | 11730717 |   0 | 43275305 |  6165890 |   0 |  1919297 |  8085187 |  3239418 |   0 |  472683 |  3712101 |   0 |   0 |   7525 | 11804813 |  79 |
|  80 | Russell        |        0 |  1744622 |   0 |  1744622 |  1027452 |   0 |   261554 |  1289006 |   436159 |   0 |   19457 |   455616 |   0 |   0 |      0 |  1744622 |  80 |
|  81 | Scott          |        0 |        0 |   0 |        0 |        0 |   0 |   539894 |   539894 |        0 |   0 |   12088 |    12088 |   0 |   0 |      0 |   551982 |  81 |
|  82 | Shenandoah     |   414879 |  6593060 |   0 |  7007939 |  3090564 |   0 |  1571850 |  4662414 |   830393 |   0 | 1515132 |  2345525 |   0 |   0 |      0 |  7007939 |  82 |
|  83 | Smyth          |   510945 |  3349610 |   0 |  3860555 |   989919 |   0 |  1119000 |  2108919 |   497795 |   0 | 1214371 |  1712166 |   0 |   0 |  39470 |  3860555 |  83 |
|  84 | Southampton    |        0 |  3929687 |   0 |  3929687 |  2314152 |   0 |   761786 |  3075938 |   484013 |   0 |  369736 |   853749 |   0 |   0 |      0 |  3929687 |  84 |
|  86 | Stafford       |        0 |        0 |   0 |        0 | 19581819 |   0 | 10186239 | 29768058 | 11496158 |   0 | 5105158 | 16601316 |   0 |   0 |      0 | 46369374 |  86 |
|  87 | Surry          |        0 |        0 |   0 |        0 |   145000 |   0 |  1020000 |  1165000 |    93446 |   0 |  839819 |   933265 |   0 |   0 |      0 |  2098265 |  87 |
|  88 | Sussex         |        0 |        0 |   0 |        0 |  1424601 |   0 |    57751 |  1482352 |   362494 |   0 |   10643 |   373137 |   0 |   0 |      0 |  1855489 |  88 |
|  89 | Tazewell       |        0 |  3106361 |   0 |  3106361 |   767501 |   0 |  1516342 |  2283843 |    74361 |   0 |  748157 |   822518 |   0 |   0 |      0 |  3106361 |  89 |
|  91 | Washington     |        0 |  3148940 |   0 |  3148940 |  1749633 |   0 |   731563 |  2481196 |   203076 |   0 |  464668 |   667744 |   0 |   0 |      0 |  3148940 |  91 |
|  92 | Westmoreland   |        0 |        0 |   0 |        0 |   295185 |   0 |   696443 |   991628 |  1230526 |   0 |  390419 |  1620945 |   0 |   0 |      0 |  2612573 |  92 |
|  93 | Wise           | 15892984 |        0 |   0 | 15892984 | 16779317 |   0 |   600744 | 17380061 |  2606874 |   0 |   72604 |  2679478 |   0 |   0 | 451864 | 20511403 |  93 |
|  94 | Wythe          |        0 |  4916109 |   0 |  4916109 |  2365156 |   0 |   961954 |  3327110 |   911959 |   0 |  667040 |  1578999 |   0 |   0 |  10000 |  4916109 |  94 |
|  95 | York           |   216786 | 11361329 |   0 | 11578115 |  4260000 |   0 |  2561895 |  6821895 |  2826453 |   0 | 1013117 |  3839570 |   0 |   0 |  13525 | 10674990 |  95 |



labeled_formulas: H=A*E
L=A*E
P=A*E
L=A*H
O=A*H
P=A*H
S=A+H
O=A*L
P=A*L
S=A+L
P=A*O
S=A+P
E=C*H
E=C*L
P=C*E
L=C*H
O=C*H
P=C*H
O=C*L
P=C*L
P=C*O
E=D*H
L=D*E
E=D*P
L=D*H
O=D*H
P=D*H
O=D*L
L=D*P
O=D*P
H=E*F
L=E*F
P=E*F
H=E*G
L=E*G
P=E*G
H=E*I
H=E*J
H=E*K
L=E*H
H=E*M
H=E*N
E=H*Q
H=E*R
H=E*S
L=E*I
P=E*I
L=E*J
P=E*J
L=E*K
E=(A*C)*H
E=(A*C)*L
P=(A+C)*E
L=(A*C)*H
O=(A*C)*H
P=(A*C)*H
S=(C*H)+A
O=(A*C)*L
P=(A*C)*L
S=(C*L)+A
P=(A+C)*O
P=(A-S)*C
E=(A*D)*H
L=(A+D)*E
E=(A*D)*P
L=(A*D)*H
O=(A*D)*H
P=(A*D)*H
S=(D*H)+A
O=(A+D)*L
L=(A*D)*P
L=(A-S)*D
O=(A*D)*P
S=(D*P)+A
H=(A*E)*F
L=(A*E)*F
P=(A*E)*F
H=(A*E)*G
L=(A*E)*G
P=(A*E)*G
H=(A*E)*I
H=(A*E)*J
H=(A*E)*K
L=(A*E)*H
H=(A*E)*M
H=(A*E)*N
E=(A*H)*Q
H=(A*E)*R
L=(A*E)*I
P=(A*E)*I
L=(A*E)*J
P=(A*E)*J
L=(A*E)*K
P=(A*E)*K
L=(A*E)*M
L=(A*E)*N
E=(A*L)*Q
L=(A*E)*R
P=(A*E)*M
P=(A*E)*N
output: {"Arithmetic-Relationship": []}
source: $MMTU_HOME/resources/Relationship-finding/AR/table_2334___xlsx_756f2de2f1b7907d0304e13012e31a03c6e2b186.xlsx___line_343