
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 the sum of the first $n$ terms of the sequence $\{a_n\}$ be $S_n$. If the inequality $a_n^2+ \frac {S_n^2}{n^2}\geqslant ma_1^2$ holds for any arithmetic sequence $\{a_n\}$ and any positive integer $n$, then the maximum value of the real number $m$ is ______.

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