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What is the best way to protect your a-pawn in a queenside castle? Is it best to slide your king over to protect the pawn and ensure the king's safety, or is it better to move the a-pawn?
The decision strongly depends on actual position. When black organizes pressure on c file it might be good to play Kb1 which protects a2 but also escapes from the dangerous file
If black pushes his queenside pawns moving a3 often creates a target and helps the opponent with pawn storming the king
On the other side, a3 may be good if the opponent attacks with his pieces rather than the pawns. It takes b4 square from him (Nb4 can be really annoying, looking at both c2 and a2)
There is no general rule, you may analyse master games and see which method they employ, especially against the structures which are more likely to occur in your games