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EXIST = 1

But AXMEN,  POINT = 0 rules out X, I, T so E is first or S is 4th

POISE

if 0 then S is not 4th so E is 1st

If 1 then S is 4th as POINT,  SNARE = 0rule out other 4 letters. 

Additionally A is 5th from EXTRA = 1 and E would be proven to not be 1st from above,  while AXMEN,  BUTTE, SNARE = 0 rule out XTR

 

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QUITE

butte-0 disproves the -u-te, quote-1 means either Q or --O-- are correct. If quite-1 then Q is proven, if it is - then --O-- is proven

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