
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 \( p(x) \) be a polynomial of degree \( 3n \) such that \( p(0) = p(3) = \cdots = p(3n) = 2 \), \( p(1) = p(4) = \cdots = p(3n-2) = 1 \), \( p(2) = p(5) = \cdots = p(3n-1) = 0 \), and \( p(3n+1) = 730 \). Find \( n \).

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