
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
Let  $m > 1$  be an integer.  A sequence  $a_1, a_2, a_3, \ldots$  is defined by  $a_1 = a_2 = 1$ ,  $a_3 = 4$ , and for all  $n \ge 4$ ,   $$ a_n = m(a_{n - 1} + a_{n - 2}) - a_{n - 3}. $$  Determine all integers  $m$  such that every term of the sequence is a square.

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