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Firstly, I've noticed your pre-flop raises are very easy to read. I always raise exactly the same amount pre-flop. So when I am raising pre-flop from late position I could have AA/KK or 44, (hopefully) no one can put me on a hand pre-flop.
Anyway, when that K hit the flop I would have probably bet into him. If he calls, I would probably put him on K-low card or two spades (obviously you'll have a better idea having played with him). I would bet to try and find out where I am in the hand. If he flat-calls he's either got an ok hand/draw to good hand. If he raises, fold - with a K on the flop you have no choice.
The way you played it is problematic; you don't know where you are in the hand on the turn. You check, he bets into you. Now he could have that K-low card or he may even have middle pair. But he could also have nothing. So you haven't learnt anything going into the turn.
Lord knows what happened on the turn. You're both scared of each other's cards. I would have, again, sent out a small bet. Maybe try and represent a Q, if he has a straight/flush draw. If he does have a straight/flush draw you don't want him to get a free card, if he raises back over you its an easy fold again, if he calls I'd probably sense that as weakness and bet the river unless it may complete a straight/flush which I would have strongly put him on.
On the river, the 3 would have scared me. In these lower-limit games it is fairly likely he has something stupid like A3s, which I haven't considered thus far. I don't like the size of the bet on the river, it seems far too large. (Though I suppose it does look like you caught three-of-a-kind 3s, but he called so he probably wasn't thinking much about your hand anyway). However you haven't found out where you are on the flop/turn, so it could be a very dangerous and expensive bet.
This is only how I would have played this. I like to play tight/agressive and constantly put my opponent under pressure to make the good decisions. You might disagree with me.
Although on a side note I have a habit of over-thinking positions like this and I'll get to the river and find out he was holding something stupid. So it's generally not necessary to think that much about most hands at the lower-limits.
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