
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
A sequence  $(a_n)$  of positive integers is defined by  $a_0=m$  and  $a_{n+1}= a_n^5 +487$  for all  $n\ge 0$ . 
Find all positive integers  $m$  such that the sequence contains the maximum possible number of perfect squares.

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