
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 \( 2n \), satisfying \( p(0) = p(2) = \cdots = p(2n) = 0 \), \( p(1) = p(3) = \cdots = p(2n-1) = 2 \), and \( p(2n+1) = -30 \). Find \( n \) and \( p(x) \).

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