
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
Find all real coefficient polynomials \( f(x) \) that satisfy the following conditions:
(1) \( f(x)=a_{0} x^{2 n}+a_{1} x^{2 n-2}+\cdots+a_{2 n-2} x^{2}+a_{2 n} \) with \( a_{0}>0 \);
(2) \( \sum_{j=0}^{n} a_{2 j} a_{2 n-2 j} \leq c_{2 n}^{n} a_{0} a_{2 n} \);
(3) The \( 2n \) roots of \( f(x) \) are all purely imaginary numbers.

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