Linecheck, bottom set 200 bb deep 3-way

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Linecheck, bottom set 200 bb deep 3-way

Blinds: $0.10/$0.25 (6 Players) SB: $67.76
BB: $89.26 (Hero)
UTG: $181.44
MP: $71.28
CO: $90.27
BN: $49.90
Cutoff is total maniac, 92/68/44 and button is loose passive, 43/7/2
Preflop ($0.35) Hero is BB with 9 9 A 6
UTG calls $0.25, MP folds, CO raises to $1.10, BN calls $1.10, SB calls $1.00, Hero calls $0.85, UTG calls $0.85
Preflop is standard I guess
Flop ($5.50) 9 A Q
SB checks, Hero checks, UTG checks, CO bets $5.27, BN calls $5.27, SB folds, Hero calls $5.27, UTG folds
PSR = 9, I hit bottom set with a blocker to AA, 3-way and board is very draw-heavy.

I didn't want to open that I hit strong as most turn cards would make it difficult for me. I also didn't have the nuts.
I also didn't want to CR, but I thought that I could check-raise any safe turn. Otherwise I could evaluate my odds at turn.
Turn ($21.31) 9 A Q 4
Hero checks, CO checks, BN bets $20.40
Safest possible turn. My plan was to CR all safe turns AI. I also have blocker to A.
Cutoff plays opens so wide that AA is unlikely and button didn't raise preflop

I think that going AI is standard here, but just want to make it sure. My hand in the end is bottom set, multiway and 200 deep in single raised pot.

Opinions?

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