
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
\[
\begin{array}{c}
p(0)=p(3)=\cdots=p(3n)=2, \\
p(1)=p(4)=\cdots=p(3n-2)=1, \\
p(2)=p(5)=\cdots=p(3n-1)=0,
\end{array}
\]
and \( p(3n+1)=730 \). Find \( n \).

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