How to best play dry A-high boards?

Lets say we open from the CU and the flop is:

A74o

Lets say my open range is:
https://pasteboard.co/J7z43Um.png

Then I will bet the flop with my value hands:
AT+, AA/77/44, A7s A4s (~15% of my range)

I will check my medium pairs:
22-KK, A2-A9 (~30% of my range)

The question is what to do with the ~55% of my range. I want to cbet this flop quite often because it forces many hands with equity against my range to fold (eg. KJ/KQ/JT etc.)

I was thinking I cbet around a third of my range that misses the flop. That way my cbet on the flop is around 50/50 value/bluffs. For the bluffs I will pick all strong backdoor drows (e.g. KJs that has a backdoor flush and streight drow).

On the turn I ll bet my bluffs again if they make a drow as well as my value hands.

On the river I ll ceck/fold most of my drows if they dont get there but mix some into my value range as bluffs.

If I check the flop my range on the turn will consist of weak top pair, 2nd pair and air. If villan bets turn and river I will generally call down with the aces, fold my hands that missed. With my middle pairs (TT-KK) I call the turn and fold the river on a 2nd bet.

What are your thoughts on that line? How many of my missed hands should I cbet on the flop? What would be a good value-bluff balance for cbets on such a board.

Thank you

The solution greatly depends upon who calls you. This board plays somewhat differently if villain is the BTN in comparison to when it's the BB.
For the latter, a solution can be found fairly quickly with GTO+ or Pio. If you go with a small size, you can bet quite a lot of "air" I think and bet a larger proportion of your range, and don't need to be quite so polarized.
I think you're more or less right about which "air" hands to pick as bluffs though. Hands with backdoors are clearly better than hands that completely whiffed. I think 3 combos of 98s would make more sense as c-bets than KJs though, since 98s folds out several hands that are already beating it, whereas if you have KJs, it's not great to fold out KTs/QJs/JT, since you dominate those hands.

Could you elaborate a bit what the GTO+ solution is if villan is in the BB?
Thank you for your answer.

I don't use solvers (I barely even play any more, so I'm not entirely sure if this is definitely a high frequency c-bet spot, or a low frequency bet) but maybe someone else will be kind enough to look up a solution to the specific COvBB situation you mentioned.

If you're really interested in finding optimal solutions, then paying the money for GTO+ will surely be worthwhile, particularly as you'll be able to tailor the input ranges according to your own player pool, and that will give you an edge on people like me that haven't done any recent work!

First of all, that opening range is fairly tight for the CO, and if your opponent in the BB isn't adjusting to your overly tight range (that is, by defending tighter himself) you can probably get away with betting small with your entire range.

But to your question about constructing a balanced checking range on this board.

Not all marginal made hands should automatically be checked. That's more of an old school way of thinking about things: "I don't want to bet my hand because it struggles to get value from worse." Solvers have shown that when you have a range advantage on the flop, lots of medium strength hands actually get played fairly aggressively. For example, KK & QQ might get checked at a high frequency on A74r, but 88 & 99 will be bet more than half the time (they need more protection from overcards). 33 and 22 will be bet nearly 100% of the time (they make for good double and triple barrel bluffs), but KQ will be checked nearly always (decent showdown value, good outs to second pair which, if made, can bluffcatch on turns and rivers). And like Arty mentioned, sometimes it also makes sense to check back some of your air that dominates some of your opponent's air, as you'll win a bigger pot when you both hit.

Don't get too caught up in value to bluff ratios on early streets. It's more about getting the frequencies right with your TOTAL range, identifying the properties of various holdings and whether betting or checking with those hands will benefit your strategy overall--not just on the flop, but setting up for turns and rivers as well.

That's a really interesting insight. Are there any other similar things that solvers have shown us, other than things like range bets capturing 99% EV etc.

Overbetting is another thing that solvers did first if I recall.