
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 fourth degree polynomials \( p(x) \) such that the following four conditions are satisfied:

(i) \( p(x) = p(-x) \) for all \( x \).

(ii) \( p(x) \geq 0 \) for all \( x \).

(iii) \( p(0) = 1 \).

(iv) \( p(x) \) has exactly two local minimum points \( x_{1} \) and \( x_{2} \) such that \( \left| x_{1} - x_{2} \right| = 2 \).

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