Turns alternate between the two players — not between the four hands. When it is your turn, you act on your two hands together. A turn has three parts:
Say you draw a tile that finishes a group in your first hand — great, keep it there. That leaves a loose tile in your second hand you no longer need, so you move a couple of concealed tiles around to tidy both racks, then discard the one tile neither hand wants. Play passes to your opponent, who now gets a chance to claim what you threw.
The rhythm — draw into a hand, rebalance, discard from a hand — repeats until someone completes both hands or the wall runs out.
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