
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  $f(x)$  which satisfy the following conditions:**i.**  $f(x) = a_0 x^{2n} + a_2 x^{2n - 2} + \cdots + a_{2n - 2}
x^2 + a_{2n}, a_0 > 0$ ;**ii.**  $\sum_{j=0}^n a_{2j} a_{2n - 2j} \leq \left(
\begin{array}{c}
2n
n\end{array} \right) a_0 a_{2n}$ ;**iii.** All the roots of  $f(x)$  are imaginary numbers with no real part.

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