Here's part of the comment on the definition of dribbling in the NFHS case book, 4-15:

"...It is not a dribble when a player stands still and holds the ball and touches it to the floor once or more than once."

So...it would appear from the comment that once A1 lets go of the ball A1 has dribbled, meaning A1 cannot pick up the ball and dribble again.

Hmmmmm Not according to 4-15-3

I took that into account with my post. I based my answer on what's in the comment on the rule. The two definitely seem contradictory but then again I'm guessing the rules makers didn't consider a scenario like this.

Setting the ball on the floor is only a problem if the player gets up in the middle of the process; because the rules makers didn't want players taking advantage of a loophole. Otherwise, I see nothing wrong with it because it doesn't create an advantage of any kind and it's not specifically disallowed. Unless I'm missing something, which is possible.

Good point, but I'm still passing on this play (if ever applicable) due to the fact that placing the ball on the floor isnt the same as pushing, throwing, batting the ball to the floor. Just my opinion.

That's cool.

Now as I play weather vane :) , the other thought that comes to mind is to travel a player has to have control. A player has control when - among other things - he/she is holding or dribbling a live ball. In the OP the player isn't doing either of those but they don't appear to have completely given up the ball, either.

Yet another hmmmmmmm.