Reviewing my latest game, I came to this point where I wasn't sure that I handled it correctly. Where do you play as black and why?

K5 from above (may be also something similar, driving White to your walls either on the right, or the left).

You have massive strength on both sides, so White can get a very cramped position on the lower edge only.

If White tries to run away, you will be able to build further influence in the centre that will radiate all over the board.

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The really most difficult Go problem ever: https://igohatsuyoron120.de/index.htm
Igo Hatsuyōron #120 (really solved by KataGo)

K5 feels very natural, but like Cassandra, I'm not sure of the exact spot.

K5 feels most natural and seems to be a good and solid move for the reasons already explained. If I would go into 'find the very best move'-mode I would go for J5. B's wall on the right seems to be bigger and more solid so I would like to push W's weak group into that direction

Nobody likes K6 or even K7? With so much massive strength on both sides, there might be no need to play as tight as K5.

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- Bantari
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WARNING: This post might contain Opinions!!

If you wanted to limit White to the lower edge, it would surely be better to play farther away from White's stone.

If you wanted to initiate a running battle, playing closer to White's stone, and probably from the left, in order to push to the right, might be preferable.

And it should be clear that a Black move on the third line is out of the question.

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The really most difficult Go problem ever: https://igohatsuyoron120.de/index.htm
Igo Hatsuyōron #120 (really solved by KataGo)

Since you might be too weak for accurate positional judgement, as W approach as far as possible just so that B cannot kill or profit well from attacking. As B attack from the outside if W has entered more deeply than the outer moyo hull (sector line). Therefore, for W, playing on the outer moyo hull is a good idea, and usually in the middle between the B walls. It is sometimes possible to play 1 intersection adjacent to that middle intersection of the outer moyo hull.

n17 is huge. If white gets it then next he can play r17. I might play that and let white jump to k5 as you can still cap and attack/kill it. black k5 could be -1 point in gote. Just because k3 has a black circle on it doesn't mean you need to play near it.

I also wanted to suggest tenuki, because it's not clear how white can live even with a second move. It is unusual to be so incredibly strong on both sides, which is why it might be that a cap is an underplay in this particular situation.

But I think that at 8k, the cap is good, reusable advice for situations somewhat like this.

Yes, when having the potential territory of both sides in mind, N17 seems to be THE move to play.

In principle, White has noteworthy territory on the upper edge only.

On the lower edge, it is not unlikely that Black will get at least as much territory as White has in both corners, even if White was allowed to play a second move there.

But:
If you ever wanted to find out what might happen after a cap (or another move nearby), the position in this game would be ideal for starting your investigations.

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The really most difficult Go problem ever: https://igohatsuyoron120.de/index.htm
Igo Hatsuyōron #120 (really solved by KataGo)

I did cap at K5, figuring that I couldn't kill the stone (it lived). My thought was to let white pick a side to escape from and profit on the other side while attacking his running group. In retrospect, I think I like J5 better since it's my weaker side.

I got N17 later. I think we both underestimated how big it was.

Thank you to everyone for the feedback and advice.

I think it would be intresting to see how W managed to live in there. At least I don't see an easy way for W to do it

Ack, I take it back. I played P17 instead of N17, because I hate points or something.

The horror show starts at move 48.

http://eidogo.com/#1YsCXubJB

Black 56:
Probably better at G5, closing the left centre.
Please remember that Go -- from its name -- is an ENCLOSING game, not a killing game.

Black 58:
G5 is a must now.
Your Black 56 has closed the right centre, so you should do the same on the left.
White will get at most about five points of territory with her group on the bottom.
This is the same as what you will get in the E5-area alone. Not to mention your potential territory in the N5-area at right.

Black 68:
Probably better Hane at G6.

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The really most difficult Go problem ever: https://igohatsuyoron120.de/index.htm
Igo Hatsuyōron #120 (really solved by KataGo)

Black 54 should be at G2. You need to understand the meaning of your thickness. White can not live by additional plays in the direction of 53. So you need not answer and have the opportunity to play from the other side. Put G2 on the board and look at the result.

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Dave Sigaty
"Short-lived are both the praiser and the praised, and rememberer and the remembered..."
- Marcus Aurelius; Meditations, VIII 21

This thinking is a major mistake. In the game situation it is absolutely necessary that you believe White can be killed. If White had invaded at K5 instead of K3, your thinking would make sense. Especially in a handicap game, it is unlikely that we could cap at K7 and kill White. However, in the game your walls are too deep. Go for the kill. Think hard about preventing White from making two eyes. If White lives, don't study how you might have done a better job on the outside, study what errors allowed such an unreasonable result. The day will soon come when your opponents will fear to play such unreasonable moves (well... except on Tygem anyway)!

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Dave Sigaty
"Short-lived are both the praiser and the praised, and rememberer and the remembered..."
- Marcus Aurelius; Meditations, VIII 21

Like ez4u said, your play after the invasion had a slack attitude. You seem to respect your opponent too much and assume he is a professional playing good moves. He is not. He is an 8k or whatever and most of his moves are rubbish*. If you can only realize which ones these are and not play as much rubbish as he does you will crush and destroy him. Here are the thoughts I would advocate (somewhat overcompensating for your timidity!):

After k3:
"Haha, you fool, too deep! You're gonna die!" or
"Haha, that move doesn't threaten either of my strong walls, I'm just gonna tenuki and play a bigger move, and then probably kill you anyway!"

After l2:
"What?! You expect me to answer that? I'm your opponent not your partner. Wake up and smell the coffee, Die Die Die!"

*Maybe rubbish is too strong for >50%, but certainly most are not optimal and at least a sizeable minority are rubbish.