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I played Black in 5 minute game, and reached this position with one minute left.

White moved its bishop close to my knight. I played knight takes knight, the white pawn took my knight. However, due to my pawns being close to the rooks, I made too many moves trying to achieve checkmate and lost on time.

[Title "Black to play"]
[FEN "2k1rr2/pp1n3p/2pb2p1/8/2BP4/P4N1P/2PB2P1/4nK2 b - - 0 1"]

1... Nxf3 {top engine move} 2. gxf3

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    You should play games with longer time controls so that you can think about these situations as they arise. The best way to learn is to figure it out for yourself. Start playing 30 minute or 60 minute games so that you can actually solve these problems at the board. Alternatively, get a physical chessboard, set the pieces up, try various moves and what you would do in response to various countermoves by white, and see what happens.
    – Alexander Woo
    Aug 23 at 20:12
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    2... Rxf3+ And you're winning the h3-pawn as well and that Rook is staying active.
    – SecretAgentMan
    Aug 23 at 22:24

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Here I would start by trading knights so both of your rooks can attack the enemy king. Since you have a huge material advantage, almost all trades will be in your favor. This means you should try to take the enemy bishops even if it's at the cost of your own pieces.

After that your opponent will be left with just king and pawns so you'll be able to take some of them and promote your own into queens.

You may want to train your technique on how to win that type of final simplified positions. For example here: https://lichess.org/practice/checkmates/piece-checkmates-i/BJy6fEDf/U3r4wkFn

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