[z10] what range to call down vs probe T+R

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[z10] what range to call down vs probe T+R

Blinds: $0.05/$0.10 (6 Players) BN: $11.70 (Hero)
SB: $7.23
BB: $9.90
UTG: $9.60
MP: $7.62
CO: $22.47
Preflop ($0.15) Hero is BN with A 4
3 folds, Hero raises to $0.25, SB folds, BB calls $0.15
Flop ($0.55) 5 T 2
BB checks, Hero checks
Turn ($0.55) 5 T 2 T
BB bets $0.30, Hero folds
Final Pot BB wins $0.53
Rake is $0.02

I am more interested in our general gameplan here than the concrete hand. In-game I was wondering what is our best hand to check behind F and able to call T + R? Even when thinking about it now I really have no clue. Do we really check-behind some Tx? They really profit from betting, to get broadways to fold, value from A high etc. I guess we start then to check-behind or worst Txs hands?

And also from the other perspective, HYPOTHETICALLY SPEAKING, can't we just probe everything + barrel R on these types of boards? Let's imagine the board runs out T52cc 8x Kx, if we just bet turn 70% and river 110%, I bet the average villain would really struggle to come up hands to call with, just as I am now?

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