Choosing a sizing in multi-way pots

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Choosing a sizing in multi-way pots

Blinds: $0.05/$0.10 (6 Players) BN: $10.32 (Hero)
SB: $12.56
BB: $13.90
UTG: $11.05
MP: $17.68
CO: $20.14
Preflop ($0.15) Hero is BN with 3 3
UTG raises to $0.30, MP calls $0.30, CO calls $0.30, Hero calls $0.30, SB folds, BB calls $0.20
Standard call having very good implied odds.
Flop ($1.55) 2 3 9
BB checks, UTG checks, MP checks, CO checks, Hero bets $0.83, BB calls $0.83, UTG folds, MP folds, CO folds
We hit our set and have a clear value bet on the flop. What are the general guidelines in that kind of spots with 3 or more players in the pot? As played should I polarize myself and bet 3/4 or even higher here? Which sizing to choose if we are for example original raiser, board is good for a range c-bet, should it also be in a range of +/- 30% of pot size or should we increase the sizing bit more?
Turn ($3.21) 2 3 9 6
BB checks, Hero bets $2.70, BB calls $2.70
I have decided to bet larger to secure a nice shove on the river and also capitalize on all draws he might have.
River ($8.61) 2 3 9 6 2
BB bets $10.07 and is all in, Hero calls $6.49 and is all in
That lead is strange and I assume he's either bluffing with missed draw or hit some kind of 2x and overplays it. Occassionally he might show up with 99. Not sure about 66 if he is able to call flop with it.

No info on any of the opponents.

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