As you described, A1 caught the ball, pivoted, and threw the ball down court.
If the ball hits the floor before A1 touches it, it's legal. If A1 does not allow the ball to touch the floor first, it is illegal.
However, would it be a travel or an illegal dribble? It seems the rules would indicate a thrown ball that is retrieved, that didn't hit the floor, would be (a) an illegal dribble, if the player's pivot foot was on the floor, and (b) a travel, if the player's pivot foot was off the floor.
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