
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 a triangle \( \triangle ABC \) with \( BC=a \), \( AC=b \), and \( AB=c \), if \( \angle A + \angle C = 2 \angle B \), find the largest positive integer \( n \) such that \( a^n + c^n \leq 2b^n \) holds for any such triangle.

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