
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
\[a_{12} x^{12} + a_{11} x^{11} + \dots + a_1 x + a_0 = 0\]
are all integers, and its roots $r_1, r_2, \dots, r_{12}$ are all integers. Furthermore, the roots of the polynomial
\[a_0 x^{12} + a_1 x^{11} + \dots + a_{11} x + a_{12} = 0\]
are also $r_1, r_2, \dots, r_{12}$. Find the number of possible multisets $S = \{r_1, r_2, \dots, r_{12}\}$.

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