
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 integers such that \(a, b, c, d, e\) are consecutive terms of an arithmetic sequence. Let \(m\) be the exact number of integer roots of the polynomial, counting multiplicity. Determine all possible values of \(m\).

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