
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
\( P(x) \) is a polynomial of degree \( 3n \) such that
\[ 
\begin{array}{l}
P(0)=P(3)=\cdots=P(3n)=2, \\
P(1)=P(4)=\cdots=P(3n-2)=1, \\
P(2)=P(5)=\cdots=P(3n-1)=0, \\
P(3n+1)=730.
\end{array}
\]

Determine \( n \).

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