So far when I've been analyzing my hands, I've just been looking at my two cards and trying to figure out what the most optimal play is. Since I want to advance my level of poker thinking to the next step, I need to try to figure out how I would play my entire range of hands given my sequence of actions and I want to work hard to make sure my lines are balanced.

In general, the whole idea (correct me if I'm wrong) is that we want to balance out the hands that we valuebet, with bluffs so we make our opponent indifferent to calling or folding. So if we are betting a polarized range on the river it will include our value hands and our air, which is the top parts and bottom parts of are range respectively. To my understanding, I think the math behind this has to do with the pot odds you offer your opponent on their call.

Example. OTR you make a pot sized bet. Your opponent since he is getting 2:1 odds (33%) and let's say all he has is a bluffcatcher so he has to have a hand that beats your hand 33% of the time. Let's say I have a range that is 50% air, then my opponent will profit by calling. But if I have a range that has 33% air and 67% value hands than no matter what choice my opponent makes it won't really matter because I basically make him so he can't profit off a call or fold against my polarized river bet. This is the premise of the whole GTO thing right?

For the middle of our range, I'm assuming that sometimes we want to be check calling, as it let's our opponent incorrectly bluff his missed draws etc and then we can get value out of the middle of our range by check calling to let our opponent bluff the bottom of his range. In this hand, I need help figuring out where my AJo lies in my range in the river after I have raised my opponents limp, bet flop, bet turn, and am deciding what the best river line is.

Opponent is loose passive. I've been raising his limps for value so I make it $1. I've been doing this with my stronger hands preflop and have been checking back my junk hands and taking the free flop.

I think I continuation bet this flop with most of my range that I raise his limp with preflop.

Once he calls on the flop, I am going to bet like this with all my strong hands (two pairs, sets, top pairs, straights) and also my good draws (flush draws, back door flush draws, pair flush draw, pair straight draws etc).

When the river sort of blanks off, I need to figure out where my top pair top kicker stands in my river range. My river range is going to consist of some of my missed flush draws, wiffed straight draws, and maybe some marginal pair draws that didn't get there. It's also going to have all my two-pairs, sets, straights, and top pair hands in it too.

How does my TPGK stand in this range? If it's in the middle of my range should I be check calling here, or is it more in the top of my range and I should valuebet it. I think since we are headsup, and villain has already shown to be "bad-passive" preflop it seems like we have an easy valuebet of 2/3 pot or so. Against a more bluff happy opponent, we might want to check call so he can bluff his wiffed draws.

Regarding sizing, another thing we could do is tailor our bet sizings to the strength of our hands. But, I want to be eventually playing bigger stakes against good opponents so I think this is bad practice since against competent players we don't want to be giving away this obvious bet sizing tells when we could just create a balanced range, and bet the same amount with our bluffs and value bets.

Sorry for the long rant, but I really want to figure out how the better players construct their ranges and get this thought process down so I can move my technical game to the next level.

Thanks!