
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
Consider a fifth-degree polynomial:
\[ x^5 + ax^4 + bx^3 + cx^2 + dx + e = 0 \]
where the coefficients are all integers. Define $m$ as the exact number of integer roots of this polynomial, counting multiplicity. Determine all possible values of $m$.

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