
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
Determine all real-coefficient polynomials \( f(x) \) that satisfy the following conditions:
(1) \(f(x)=a_{0} x^{2 n}+a_{2} x^{2 n-2}+\cdots+a_{2 n-2} x^{2}+a_{2 n}, a_{0}>0\);
(2) \(\sum_{j=0}^{n} a_{2 j} a_{2 n-2 j} \leqslant \binom{2 n}{n} \cdot a_{0} a_{2 n}\);
(3) \(f(x)\) has \(2 n\) roots, all of which are purely imaginary numbers.

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