
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
Anthony writes the  $(n+1)^2$  distinct positive integer divisors of  $10^n$ , each once, on a whiteboard. On a move, he may choose any two distinct numbers  $a$  and  $b$  on the board, erase them both, and write  $\gcd(a, b)$  twice. Anthony keeps making moves until all of the numbers on the board are the same. Find the minimum possible number of moves Anthony could have made.

*Proposed by Andrew Wen*

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