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You CAN flat call preflop. And I'll explain why in a moment. If you do, your play postflop is to shove. No matter what comes, you shove. It's called a stop-and-go, and you are in a decent spot to do it. In fact, with 44-77, as well as some Ax hands, this might be my preferred line.
Here's things you need to look at when considering a stop-and-go:
1. Position - You MUST be out of position postflop for this to be an option
2. Stack size - You need enough chips to call a bet and make your opponent fold to a shove on the flop, but not so much that he is likely to fold most hands to a resteal preflop.
3. Strength of your hand - the stronger your hand, the more likely you should be to push over the top. That said, there are a lot of hands which are likely slightly ahead preflop, such as modest pocket pairs and Ax holdings (the latter mostly against late position players). With these hands, you are the likely favorite, but not necessarily enough so that you want to push when you are likely to be called. You are likely to be even more ahead once the flop has come, since your opponent will usually miss the flop (thus coinflips become 65/35).
4. Knowledge of your opponent - does he tend to raise from the position he is in with a lot of the hands that you want to be against in these situations? Specifically, will he either raise any two broadway and/or suited connectors, or even any two cards? That argues for making the play. Also, does he tend to call tightly or lightly when he misses a flop? That will be a big factor as well.
Here, our position is perfect. Our stack is ok - though we do have enough to shove over and get mostly folds. Our hand is ok - strong enough to make the play, certainly, but also strong enough to shove over, as discussed above.
I could go either way on whether to call or to shove. The determining factor is likely to be how active my opponent has been.
Again, if you do call, your play is absolutely to shove ANY flop.
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