Imagine you're playing a game of crazyhouse, which operates like standard chess, except that you have a pocket of captured pieces that you can drop on the board in any legal location as your move.
This is the current state of the board with Black to move.
Here are the pieces in White's hand:
Your task: Find the minimum total-value piece combination Black must have in their pocket to force a checkmate.
Assume pawns are worth 1 point, knights and bishops are worth 3, rooks are worth 5, and queens are worth 9. The pieces in your pocket do not need to be legal (e.g. they don't depend on you actually having taken pieces off the board).
The checkmark will go to the player who finds the lowest total. Because this puzzle is trivially easy with a chess engine, I ask that participants refrain from using an engine. My total (which I'll reveal in a week), was found by manually playing out a game in an analysis board WITHOUT engine assistance.