
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
\[x^5 + ax^4 + bx^3 + cx^2 + dx + e = 0\] are all integers. Let $n$ be the exact number of integer roots of the polynomial, counting multiplicity. For example, the polynomial $(x - 1)^3(x + 2)^2 = 0$ has five integer roots counting multiplicity, as root $-1$ is counted three times and $2$ is counted twice.

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

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