
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
Given that \( a \), \( b \), and \( c \) are the lengths of the sides of a right triangle, and for any positive integer \( n \) greater than 2, the following equation holds: 
\[ \left(a^{n} + b^{n} + c^{n}\right)^{2} = 2\left(a^{2n} + b^{2n} + c^{2n}\right) \]
Find \( n \).

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