
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
The coefficients of the polynomial
\[x^5 + bx^4 + cx^3 + dx^2 + ex + f = 0\]are all integers. Let $n$ be the exact number of integer roots of the polynomial, counting multiplicity. For example, the polynomial $(x + 4)^3 (x^2 + x + 5) = 0$ has three integer roots counting multiplicity, because the root $-4$ is counted thrice.

Enter all possible values of $n,$ separated by commas.

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