
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
Find the smallest positive integer \( n \) such that the square mean of the first \( n \) natural numbers (\( n > 1 \)) is an integer. The square mean of \( n \) numbers \( a_1, a_2, \cdots, a_n \) is defined as
$$
\left(\frac{a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2}}{n}\right)^{\frac{1}{2}} .
$$

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