***Getting KK cracked by A-pair on flop in LP in early levels of a tou

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My KK's have been getting cracked off late. I was wondering if I could understand the prob of this happening.

***Getting KK cracked by A-pair on flop in LP in early levels of a tourney.***

PART 1. I have KK, I bet 4X BB PF and get called by a player who's played solidly so far. Everybody else folds. Flop comes A6T. I bet 1/2Pot, he calls. What is the prob he has an A?

My take:

I put him on AK-to-AT (as he is in LP) or QQ-99. (I personally would ideally re-raise my QQ-99 80% of the time.) I believe these are also hands that he would call a 1/2pot c-bet with.

So he has 3*14 = 56 (3aces, 14 remaining cards out of (K,Q,J,T)) overcards which beat us.
And he has 4*6 = 24 pair cards which we beat.

Ace-card probability:
=56/80 = 70% . I adjust that to 2/3 to accomodate some bluffing possibilities.

So if he wins with 2:1, I have concluded I need to be a bit bold and bet the pot or more and try to take the pot right away. If he calls, I let the pot go on further streets.

PART 2.

So what is my strategy when an A-rag-rag falls on the flop as in the above scenario?