
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 
\[x^3 + bx^2 + cx + d = 0\]
are all integers. Let $m$ be the exact number of integer roots of this polynomial, counting multiplicity. For example, the polynomial $(x + 2)^2 (x - 3) = 0$ has three integer roots counting multiplicity, because the root $+2$ is counted twice, and $-3$ is counted once.

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

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