
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 polynomial of the form
\[x^4 + ax^3 + bx^2 + cx + d = 0\]
where all coefficients are integers. Determine all possible values of $m$, the exact number of distinct integer roots of the polynomial, counting multiplicity. For example, the polynomial $(x - 2)^2(x^2 + x + 1) = 0$ has one distinct integer root, $2$, counted twice.

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