
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) $f(x)=a_{0} x^{2n}+a_{2} x^{2n-2}+\cdots+a_{2n-2} x^{2}+a_{2n}$ with $a_{0}>0$;
(2) $\sum_{j=0}^{n} a_{2j} \cdot a_{2n-2j} \leqslant C_{2n}^{n} a_{0} a_{2n}$;
(3) All $2n$ roots of $f(x)$ are purely imaginary numbers.

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