
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  $n\geq 2$  be a positive integer. Find the minimum value of positive integer  $m$  for which there exist positive integers  $a_1,\ a_2,\ \cdots, a_n$  such that : $\bullet\ a_1<a_2<\cdots <a_n=m$  $\bullet \ \frac{a_1^2+a_2^2}{2},\ \frac{a_2^2+a_3^2}{2},\ \cdots,\ \frac{a_{n-1}^2+a_n^2}{2}$  are all square numbers.

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