Cho Chikun Life&Death Puzzle 45

I am currently doing Cho Chikun Life and Death puzzles by sunspark

I have a problem understnading how Puzzle 45 works.
I have followed all the hints, reached the final solution and still cant figure out how the group can still be alive.
I tried further moves to test. And still the group can be captured.
Can someone help explaining? Thanks!

The crucial move here is white Q2. To avoid that white creates an eye over there with R1, black has to play R1. This gives white the opportunity to play T4.
White is now alive because T1 and T 2 are an eye, while black cannot play R2 (to falsify the eye at S3) because that is a self atari.
To prevent the self atari black has to play a stone at P1 first, and this gives white the opportunity to play R2. The white group lives.

I didn’t even need to open this thread to know which problem this was, it has been asked about several times on multiple forums! (e.g. https://www.reddit.com/r/baduk/comments/ji9f08/smashing_my_head_against_this_help/) It’s a blindspot for many (myself included, I recall getting stuck on it years ago). The crucial thing with the correct move 3 is it prevents the eye becoming false in sente, but it’s an unintuitive empty triangle.

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