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:
|   K | L                   |            M |         N |            O |    P |      Q |            R |         S |          T |           U |            V |
|----:|:--------------------|-------------:|----------:|-------------:|-----:|-------:|-------------:|----------:|-----------:|------------:|-------------:|
|  10 | Alamance County     |   9764747653 | 158565587 |   9606182066 | 2009 | 1.0001 |   9605221544 | 158565587 |  260006341 |  1946452123 |  11970245595 |
|  20 | Alexander County    |   2127294728 | 129246201 |   1998048527 | 2007 | 0.921  |   2169433797 | 129246201 |   66011499 |   364599716 |   2729291213 |
|  30 | Alleghany County    |   1590096379 | 167535800 |   1422560579 | 2007 | 0.8982 |   1583790446 | 167535800 |   36020775 |   165647363 |   1952994384 |
|  50 | Ashe County         |   3366709350 |  87278700 |   3279430650 | 2006 | 0.7809 |   4199552632 |  87278700 |   63117304 |   332132639 |   4682081275 |
|  60 | Avery County        |   4271112199 |  68438500 |   4202673699 | 2010 | 0.933  |   4504473418 |  68438500 |   32774494 |   247065632 |   4852752044 |
|  70 | Beaufort County     |   3995028395 | 253787775 |   3741240620 | 2010 | 0.9894 |   3781322640 | 253787775 |   83262357 |  1534011860 |   5652384632 |
|  90 | Bladen County       |   1904711008 | 162092959 |   1742618049 | 2007 | 0.8498 |   2050621380 | 162092959 |   86187515 |   591620475 |   2890522329 |
| 100 | Brunswick County    |  30860536686 |  65678640 |  30794858046 | 2007 | 1.0712 |  28748000416 |  65678640 | 1153728963 |  1477500607 |  31444908626 |
| 110 | Buncombe County     |  25454061649 | 335053000 |  25119008649 | 2006 | 0.8955 |  28050260915 | 335053000 |  513573883 |  3106994363 |  32005882161 |
| 120 | Burke County        |   5381184035 |  76840191 |   5304343844 | 2007 | 0.9775 |   5426438715 |  76840191 |  181207434 |  1127897722 |   6812384062 |
| 130 | Cabarrus County     |  18231055772 |  90023400 |  18141032372 | 2008 | 1.017  |  17837789943 |  90023400 |  271487306 |  2713076492 |  20912377141 |
| 140 | Caldwell County     |   4425153092 |  72542800 |   4352610292 | 2005 | 0.887  |   4907114196 |  72542800 |  150835535 |   876161453 |   6006653984 |
| 150 | Camden County       |   1034079691 |  60905926 |    973173765 | 2007 | 1.0264 |    948142795 |  60905926 |   15580274 |   194948014 |   1219577009 |
| 160 | Carteret County     |  18228589564 |  69952621 |  18158636943 | 2007 | 1.0825 |  16774722349 |  69952621 |  140251857 |   947602402 |  17932529229 |
| 180 | Catawba County      |  12169956713 |  77831500 |  12092125213 | 2007 | 0.9772 |  12374258302 |  77831500 |  610253127 |  2101925651 |  15164268580 |
| 190 | Chatham County      |   7428432828 | 347026393 |   7081406435 | 2009 | 1.0003 |   7079282650 | 347026393 |  250159004 |   957230406 |   8633698453 |
| 200 | Cherokee County     |   3756979132 |  80996140 |   3675982992 | 2008 | 1.0754 |   3418247156 |  80996140 |   53931528 |   287313305 |   3840488129 |
| 210 | Chowan County       |   1244622970 |  89083045 |   1155539925 | 2006 | 0.8992 |   1285075539 |  89083045 |   22724762 |   196846953 |   1593730299 |
| 220 | Clay County         |   1932098374 |  40560300 |   1891538074 | 2010 | 1      |   1891538074 |  40560300 |   27322790 |   126573752 |   2085994916 |
| 230 | Cleveland County    |   5005899996 | 201871766 |   4804028230 | 2008 | 0.972  |   4942415874 | 201871766 |  374726300 |  1268395519 |   6787409459 |
| 240 | Columbus County     |   2251404599 | 221531400 |   2029873199 | 2005 | 0.8442 |   2404493247 | 221531400 |  120606017 |   984263308 |   3730893972 |
| 250 | Craven County       |   8035423356 | 113839891 |   7921583465 | 2010 | 0.999  |   7929512978 | 113839891 |  141105952 |  1416600179 |   9601059000 |
| 260 | Cumberland County   |  17364299262 |  78986015 |  17285313247 | 2009 | 0.9986 |  17309546612 |  78986015 |  348880372 |  3139953308 |  20877366307 |
| 270 | Currituck County    |   7821720991 |  81517376 |   7740203615 | 2005 | 0.977  |   7922419258 |  81517376 |   73394050 |   367736289 |   8445066973 |
| 300 | Davie County        |   3561644075 |  53499030 |   3508145045 | 2009 | 1.0172 |   3448825251 |  53499030 |   72921323 |   609313298 |   4184558902 |
| 310 | Duplin County       |   2867080898 | 200639700 |   2666441198 | 2009 | 0.9885 |   2697462011 | 200639700 |  114493625 |   727388274 |   3739983610 |
| 320 | Durham County       |  24366514483 |  58939635 |  24307574848 | 2008 | 0.9994 |  24322168149 |  58939635 |  504009682 |  4641220622 |  29526338088 |
| 330 | Edgecombe County    |   2282424718 | 196787543 |   2085637175 | 2009 | 0.9877 |   2111609978 | 196787543 |  144975889 |   653563295 |   3106936705 |
| 340 | Forsyth County      |  27991856671 | 109213778 |  27882642893 | 2009 | 1.0041 |  27768790851 | 109213778 |  606245905 |  5303728778 |  33787979312 |
| 350 | Franklin County     |   3243365470 | 172387286 |   3070978184 | 2004 | 0.9012 |   3407654443 | 172387286 |  126014026 |   667884914 |   4373940669 |
| 360 | Gaston County       |  11599501504 |  97704955 |  11501796549 | 2007 | 0.9633 |  11939994341 |  97704955 |  703877245 |  2327213202 |  15068789743 |
| 390 | Granville County    |   3134588764 | 208039503 |   2926549261 | 2010 | 0.9822 |   2979585890 | 208039503 |  113672985 |   699405583 |   4000703961 |
| 410 | Guilford County     |  36806720820 |  52025934 |  36754694886 | 2004 | 0.9313 |  39466009756 |  52025934 | 1035538236 |  7034889502 |  47588463428 |
| 420 | Halifax County      |   2659342572 | 142140284 |   2517202288 | 2007 | 0.9148 |   2751642204 | 142140284 |  166430768 |   724541218 |   3784754474 |
| 430 | Harnett County      |   5842517243 | 161577720 |   5680939523 | 2009 | 0.9983 |   5690613566 | 161577720 |  167779881 |  1024270087 |   7044241254 |
| 440 | Haywood County      |   6252182637 | 167571354 |   6084611283 | 2006 | 0.8636 |   7045636039 | 167571354 |  132236474 |   824087904 |   8169531771 |
| 450 | Henderson County    |  11437247363 | 158081190 |  11279166173 | 2007 | 0.9156 |  12318879612 | 158081190 |  201935623 |  1587248217 |  14266144642 |
| 470 | Hoke County         |   2008707025 |  80883150 |   1927823875 | 2006 | 0.9299 |   2073151817 |  80883150 |   63334616 |   471945314 |   2689314897 |
| 480 | Hyde County         |   1023739903 | 108611412 |    915128491 | 2009 | 0.976  |    937631651 | 108611412 |   23280343 |    76369064 |   1145892470 |
| 490 | Iredell County      |  17283957375 | 299201060 |  16984756315 | 2007 | 0.96   |  17692454495 | 299201060 |  329279232 |  3083744281 |  21404679068 |
| 500 | Jackson County      |  10675216097 | 145520764 |  10529695333 | 2008 | 1.058  |   9952453056 | 145520764 |  151137486 |   422724532 |  10671835838 |
| 510 | Johnston County     |   9420344490 | 214285240 |   9206059250 | 2003 | 0.8813 |  10445999376 | 214285240 |  267414070 |  2117600813 |  13045299499 |
| 530 | Lee County          |   3460008982 |  46821700 |   3413187282 | 2007 | 0.9538 |   3578514659 |  46821700 |  119918246 |  1221752061 |   4967006666 |
| 540 | Lenoir County       |   2698818023 | 186891559 |   2511926464 | 2009 | 1.0023 |   2506162291 | 186891559 |   93262570 |   868192079 |   3654508499 |
| 550 | Lincoln County      |   7097275375 | 128000523 |   6969274852 | 2008 | 1.0255 |   6795977428 | 128000523 |  346324378 |  1165645857 |   8435948186 |
| 560 | Macon County        |   8765547397 | 108286860 |   8657260537 | 2007 | 1.0335 |   8376642997 | 108286860 |  132650817 |   388254895 |   9005835569 |
| 570 | Madison County      |   1661871080 | 118997600 |   1542873480 | 2004 | 0.7075 |   2180739901 | 118997600 |   49327292 |   204126330 |   2553191123 |
| 600 | Mecklenburg County  |  82567397358 |  69987887 |  82497409471 | 2003 | 0.8558 |  96398001251 |  69987887 | 3226935045 | 14382114488 | 114077038671 |
| 610 | Mitchell County     |   1514681384 |  58930000 |   1455751384 | 2009 | 1.0037 |   1450384960 |  58930000 |   66481926 |   235227814 |   1811024700 |
| 620 | Montgomery County   |   1830631650 |  52618200 |   1778013450 | 2004 | 0.7389 |   2406297808 |  52618200 |   59913211 |   429480284 |   2948309503 |
| 630 | Moore County        |  10569175241 | 276502070 |  10292673171 | 2007 | 0.952  |  10811631482 | 276502070 |  154515305 |  1015565163 |  12258214020 |
| 640 | Nash County         |   5289888120 | 215751853 |   5074136267 | 2009 | 0.9779 |   5188808945 | 215751853 |  114209989 |  1487912911 |   7006683698 |
| 650 | New Hanover County  |  30275232455 |  33891598 |  30241340857 | 2007 | 1.0023 |  30171945383 |  33891598 |  429491375 |  3085387490 |  33720715846 |
| 660 | Northampton County  |   1482861266 | 173248656 |   1309612610 | 2007 | 0.8727 |   1500644677 | 173248656 |  100533820 |   278289306 |   2052716459 |
| 670 | Onslow County       |  10566625687 | 146074040 |  10420551647 | 2010 | 0.9752 |  10685553371 | 146074040 |  221310911 |  1495899588 |  12548837910 |
| 680 | Orange County       |  14294533401 | 282905006 |  14011628395 | 2009 | 0.9876 |  14187554065 | 282905006 |  229583067 |  1157628659 |  15857670797 |
| 690 | Pamlico County      |   1195705916 |  42013863 |   1153692053 | 2004 | 0.587  |   1965403838 |  42013863 |   17808691 |   185820961 |   2211047353 |
| 700 | Pasquotank County   |   2881332197 | 102484500 |   2778847697 | 2006 | 0.9198 |   3021143397 | 102484500 |   52532359 |   396566388 |   3572726644 |
| 710 | Pender County       |   4065315004 |  85063888 |   3980251116 | 2003 | 0.6586 |   6043503061 |  85063888 |   69855719 |   641078729 |   6839501397 |
| 740 | Pitt County         |   9457203613 | 244368788 |   9212834825 | 2008 | 0.9858 |   9345541515 | 244368788 |  105153588 |  2046502463 |  11741566354 |
| 750 | Polk County         |   2466501199 |  96865492 |   2369635707 | 2009 | 0.8989 |   2636150525 |  96865492 |   64303715 |   210370241 |   3007689973 |
| 760 | Randolph County     |   7942292364 | 125904069 |   7816388295 | 2007 | 0.9661 |   8090661728 | 125904069 |  248750002 |  1981939059 |  10447254858 |
| 780 | Robeson County      |   4132684273 | 222706600 |   3909977673 | 2010 | 0.9847 |   3970729840 | 222706600 |  261979822 |  1414415472 |   5869831734 |
| 790 | Rockingham County   |   4394657051 | 141378962 |   4253278089 | 2003 | 0.8986 |   4733227341 | 141378962 |  460584600 |  1178942604 |   6514133507 |
| 800 | Rowan County        |   9204040991 | 280093817 |   8923947174 | 2007 | 0.9724 |   9177238970 | 280093817 |  496178719 |  2141841443 |  12095352949 |
| 810 | Rutherford County   |   4921420030 |  91430630 |   4829989400 | 2007 | 0.9261 |   5215408055 |  91430630 |  316644030 |   681168900 |   6304651615 |
| 820 | Sampson County      |   2640711741 | 486042644 |   2154669097 | 2003 | 0.8128 |   2650921625 | 486042644 |  120004198 |   698477186 |   3955445653 |
| 830 | Scotland County     |   1317127174 |  61012160 |   1256115014 | 2003 | 0.9142 |   1374004609 |  61012160 |   74451218 |   507361041 |   2016829028 |
| 840 | Stanly County       |   3417157656 |  72986806 |   3344170850 | 2005 | 0.8676 |   3854507665 |  72986806 |   96045649 |   717886926 |   4741427046 |
| 860 | Surry County        |   4186229083 | 230787230 |   3955441853 | 2008 | 0.9668 |   4091272086 | 230787230 |  150474702 |   951252214 |   5423786232 |
| 880 | Transylvania County |   5472244297 |  30117670 |   5442126627 | 2009 | 0.9949 |   5470023748 |  30117670 |  107930454 |   384606683 |   5992678555 |
| 900 | Union County        |  19860204041 | 406299116 |  19453904925 | 2008 | 1.0131 |  19202354086 | 406299116 |  357989385 |  2823929518 |  22790572105 |
| 920 | Wake County         | 103067246104 | 362647492 | 102704598612 | 2008 | 1.0149 | 101196766787 | 362647492 | 2837941846 | 13395143774 | 117792499899 |
| 930 | Warren County       |   2267774111 |  96862917 |   2170911194 | 2009 | 1.0193 |   2129805939 |  96862917 |   56533179 |   199622303 |   2482824338 |
| 950 | Watauga County      |   8195293782 | 111993356 |   8083300426 | 2006 | 0.8482 |   9529946270 | 111993356 |   79396684 |   481131822 |  10202468132 |
| 970 | Wilkes County       |   4579256321 |  66494430 |   4512761891 | 2007 | 0.9522 |   4739300453 |  66494430 |  125284658 |   806067396 |   5737146937 |
| 980 | Wilson County       |   4688131501 | 173030151 |   4515101350 | 2008 | 0.998  |   4524149649 | 173030151 |   78218823 |  1614575985 |   6389974608 |
| 990 | Yadkin County       |   2267813602 | 240790818 |   2027022784 | 2009 | 1.0189 |   1989422695 | 240790818 |   64127478 |   447546829 |   2741887820 |
| 995 | Yancey County       |   2361726133 |  81918300 |   2279807833 | 2008 | 0.971  |   2347896841 |  81918300 |   44888637 |   213936740 |   2688640518 |



labeled_formulas: S=(K/M)+N
S=(K/O)+N
S=(Q/K)+N
S=(K/R)+N
S=(K/T)+N
S=(K/U)+N
S=(K/V)+N
M=(N+O)+Q
S=(P/M)+N
S=(Q/M)+N
S=(T/M)+N
M=(O+Q)+S
S=(P/O)+N
S=(Q/O)+N
S=(T/O)+N
S=(Q/P)+N
S=(P/R)+N
S=(P/T)+N
S=(P/U)+N
S=(P/V)+N
S=(Q/R)+N
S=(Q/T)+N
S=(Q/U)+N
S=(Q/V)+N
S=(T/R)+N
S=(T/V)+N
output: {"Arithmetic-Relationship": []}
source: $MMTU_HOME/resources/Relationship-finding/AR/table_633464___xlsx_07d21d936997b9921df4d02d1f9271edbf5b5353.xlsx___line_34