
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 \( 2n \) such that:
\[
\begin{array}{l}
P(0) = P(2) = \cdots = P(2n) = 0, \\
P(1) = P(3) = \cdots = P(2n-1) = 2, \\
P(2n+1) = -6.
\end{array}
\]

Determine \( n \) and \( P(x) \).

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