
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) \( f(x) = a_{0} x^{2n} + a_{2} x^{2n-2} + \cdots + a_{2n-2} x^{2} + a_{2n} \), \( a_{0} > 0 \);
(2) \( \sum_{j=0}^{n} a_{2j} a_{2n-2j} \leq c_{2n}^{n} \cdot a_{0} a_{2n} \);
(3) The \( 2n \) roots of \( f(x) \) are all purely imaginary numbers.
(1997 China National Team Selection Contest)

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