
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 polynomial  $ p(x)$  such that the following four conditions are satisfied:
(i)  $ p(x)\equal{}p(\minus{}x)$  for all  $ x$ ,
(ii)  $ p(x)\ge0$  for all  $ x$ ,
(iii)  $ p(0)\equal{}1$ 
(iv)  $ p(x)$  has exactly two local minimum points  $ x_1$  and  $ x_2$  such that  $ |x_1\minus{}x_2|\equal{}2$ .

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