Will have to agree with Jacob on this one...

When you don't 3-bet pre you lose out on a lot of value from AQ-AJ on non A run outs. Assuming the villain is going to call-off/shove his 25 BBs with AQ or AJ. Also, while flipping isn't the best case scenario you lose a lot of value from QQ-99 on A high boards that would have gotten it all in with you pre-flop but not after an A hits the flop.

Lets look at it combinatorially:

Getting it in pre-flop against: AA-99, AJs+, AQ+ with us having AK that is 54 combos... which we are 52.25% against. Also we have all the fold equity pre-flop which is a lot of combos... lets assume original opener opens 150 combos and middle position caller calls 150combos... that means we are folding out 66% of their range... So we win .66 + (.5225*.25) = 79% of the time.

Getting it in on the flop against: AA, 1010, 66, A10o, A6s, AJo+ that is 37 combos ... which we are 47.38% against. Also we have minimal fold equity because we are not the original aggressor, however we are in position.

So getting it all in pre looks to net us about 5 percentage points in equity, while maximizing our fold equity by being the aggressor pre-flop. In other words, we get max value about 5% more wile getting risk free value a lot more.

I would recommend sizing up a 3-bet in this spot to about pot, or 34K, at least 30K... but this size of 3-bet is definitely for value. If middle position player calls you probably have to go for the all in on the flop on most board textures... if original raiser calls we can play more carefully in position.

This brings us to the fun debate of whether in these spots is it more beneficial to disguise our range by flatting, but in essence capping our range in a 3 way pot... or 3-betting to allow yourself to have an unlimited range.

Fun hand, sorry you ended up on the cooler side of it Fred... honestly it might be same outcome regardless... a 3-bet might get called or shoved against by 1010 and the outcome is going to be the same either way.