
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 polynomials  $P(X)$  of fourth degree with real coefficients that verify the properties:**a)**  $P(-x)=P(x), \forall x\in\mathbb{R};$ **b)**  $P(x)\geq0, \forall x\in\mathbb{R};$ **c)**  $P(0)=1;$ **d)**  $P(X)$  has exactly two local minimums  $x_1$  and  $x_2$  such that  $|x_1-x_2|=2.$ 

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