
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 mean square of \( n \) numbers \( a_{1}, a_{2}, \cdots, a_{n} \) be defined as \(\left(\frac{a_{1}^{2} + a_{2}^{2} + \cdots + a_{n}^{2}}{n}\right)^{\frac{1}{2}}\). Let \( M \) be the set of all values of \( n \) such that the mean square of the first \( n \) positive integers is an integer, where \( n > 1 \). Find the smallest element in the set \( M \).

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