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The thing about whether or not it would tick down at the end of your turn or the beginning does not matter for whether or not you would be able to use it the following turn. If it ticks down at the beginning of your turn and you rolled a one, your next turn begins, that ticks to 0 and you can use the breath weapon again.
However, if it ticks down at the end of your turn and you roll a one, then your CURRENT turn ends and then when your next turn begins you can use it.
No matter if it’s at the beginning or the end, you would be able to use the ability on your next turn.