
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 mean of the squares of the first \( n \) natural numbers (\( n > 1 \)) is an integer.

(Note: The mean of the squares of \( n \) numbers \( a_1, a_2, \cdots, a_n \) is given by \( \sqrt{\frac{a_{1}^2 + a_{2}^2 + \cdots + a_{n}^2}{n}} \).)

(Note: Fifteenth American Mathematical Olympiad, 1986)

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