At first, on a board, the number $1$ is written $100$ times. Every minute, we pick a number $a$ from the board, erase it, and write $\frac{a}{3}$ thrice instead. We say that a positive integer $n$ is persistent if after any amount of time, regardless of the numbers we pick, we can find at least $n$ equal numbers on the board. Find the greatest persistent number.

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