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I don't understand Puzzle 63650

Why does White move his Queen to e3 when it results in losing it? Compared to moving it away to a safe square of which there are a few choices. I failed this puzzle and calculated the lines correctly, but I didn't take into account that e3 was a valid contender because of the above. Am slightly miffed.
de.lichess.org/training/63650

The answer is that after Qd3, c2 or b1 black plays Rh4, threatening the skewer Rh1+ winning the rook on a1. I would not see all variations in that human defences by white, but at least that line looks very dangerous for the white king. I maybe would have given that move a try. The rating is over 2100, therefore that's not only a problem for you.
I'm curious what first move you (black) would otherwise make in that puzzle?
@KateKidna I really struggled to see one. I spent ages on this puzzle and couldn't find a way forward for Black. Black starts down material so any exchange, looked unfavorable to me, especially Bf5, which had a good pin on the Queen. The move I tried was Qh4, double-attacking the Knight.
I solved the problem because white played the rather weak Qe3 allowing Rf1+. If Qd3 Rh4 I imagine there might be a possible forced win for black but I can't calculate it if looking briefly only.
I actually managed to solve it first try (without looking at all your comments here!) I guess I got lucky or something.
Thanks , I think I solved it more through luck than anything else now!
What made you think that taking the Knight was a good idea though, given you start at a material disadvantage? Isn't it the idea to avoid simplifying in that case?
After your hint in the first post i tried to find any way attacking the queen. Without any hint i would have done it the right way only by getting frustrated because i can't find anything and move my best guess.

Black opens the f-file, maybe getting something by doubling there.
Maybe you dream about rook on h4, the other one on f8 and then Qe3 Rf2 Qxf2# The white king has not many squares.

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