I've looked at it for 15 minutes without a good answer.
Exercice : play a lost position

It's an hard position.
In theory the white has the queen for the rook, so I think first at the defence if I don't see good defensive options, I see if possible to attack the black position

Yes, I looked to defend and I didn't see much. I then looked to cause trouble for black and focused on white's more powerful piece, the queen, to start a problem attack, but could not see any...

Yes, the only chance for the white now, is weaken the black castle.
But is interesting, to think at the variants.

I'll have a try at some variations after 1.e6, although I am sure I am missing a lot of stuff, and I hope you can correct me. I am not moving pieces on the board since I think that is the point of the exercise. That means I might missing something totally trivial at any point, sorry for wastig your time in case:)
After 1. e6 I considered 1...fxe, 1...f6 and 1...Rxd2
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1...fxe6 2. Qxe6+ Kh8 3. Bd5 and I prevent the rook from eating d2, and ...Bxc2 is also not possible due to checkmate in g8. After 3...Bc5 maybe I could just play 4. Qe5 or 4. Qc6
Actually I also considered 1...fxe6 2. Bd5 exd5 (2...Rxd5 3. Qe6+ and 4. Qxd5) 3. Qe6+ and 4. Qe1, but I dont't think it is necessary, although I thought it was cute.
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1...f6 2. e7 Bxe7 3. Qe6+ Kf8 4. Bd5 and I think whit eis even winning here.. If 3...Kh8 I think I want to play 4.Bd5 anyway and not take in e7
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1...Rxd2 2. exf7+ Kxf7 3. Qc4+ (I thought this could be best, to prevent black K from going back to g8, but I am really lost with all the possibilities here, so I am not sure...) and now
3...Kg6 4. Be4+ and i see a lot of checks coming, although not sure if I have a perpetual
3...Ke8 4. Qe6+ Be7 (4...Kd8 5. Qb6+) and I don't see much else I can do here with white, 5. Qc8+ Rd8 or 5. Bc6+ Rf8 and I really can't see a way to save it...
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Oh well, it was a mess of a position and I am sure I miscalculated a lot, but it was fun trying. Thanks Alex for the exercise.
PS Actually loking at it again I think that also 1...Rxd2 2. exf7 Rh8 is strog for black, I find it really hard to calculate everything, really tough position for me...
An important skill of a strong players, is find new resources also in lost positions. So, I think we can do an useful exercise , think about this position.
It's the white turn, the black threatens the promotion and it seems impossible stop this.
Think 15 minutes at this position and find an idea to play!
Remind that the position is lost, but let's try to find an idea that can make problems at the black position.