
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
For a polynomial  $P(x)$  with integer coefficients and a prime  $p$ , if there is no  $n \in \mathbb{Z}$  such that  $p|P(n)$ , we say that polynomial  $P$  *excludes*  $p$ . Is there a polynomial with integer coefficients such that having degree of 5, excluding exactly one prime and not having a rational root?

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