
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 the number of polynomials \( p(x) \) with integer coefficients satisfying \( p(x) \geq \min \left\{2 x^{4}-6 x^{2}+1,4-5 x^{2}\right\} \) and \( p(x) \leq \max \left\{2 x^{4}-6 x^{2}+1,4-5 x^{2}\right\} \) for all \( x \in \mathbb{R} \).

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