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I would be considering 9. b3 or trying to find another good waiting move, like a rook move. There doesn't seem any need for a4. Let black advance the pawns there. White wants to wait til black castles 0-0 and play Nd5 followed by Ne3. Black is not in a good position.
Why is 9. a4 the best move? What's behind that move?
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Nd5 wins a queen or a rook, that should be best. the queen's only good square is qa4, which is followed up by nc7+. i dont know why stockfish suggests a4
Last time I checked the rules, Queens could move backwards. So, 9.Nd5 Qd8 suggests itself.