Jonathan has a magical coin machine that accepts coins in amounts of $7$, $8$, and $9$. If he puts in $7$ coins, he gets $3$ coins back; if he puts in $8$, he gets $11$ back; and if he puts in $9$, he gets $4$ back. The coin machine does not allow two entries of the same amount to happen consecutively. Starting with $15$ coins, what is the minimum number of entries he can make to end up with $4$ coins?

Let's think step by step and output the final answer within \boxed{}.
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