
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
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  $a/3$  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.

Present the answer in LaTex format: \boxed{Your answer}
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