
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 the polynomial
\[ p(x) = a_8 x^8 + a_7 x^7 + a_6 x^6 + \dots + a_2 x^2 + a_1 x + a_0 \]
where all coefficients \(a_i\) are integers and its roots \(r_1, r_2, \dots, r_8\) are all integers or complex numbers. Furthermore, the roots of the polynomial
\[ q(x) = a_0 x^8 + a_1 x^7 + a_2 x^6 + \dots + a_6 x^2 + a_7 x + a_8 \]
are the same as those of \(p(x)\). Determine the number of possible multisets of roots \(S = \{r_1, r_2, \dots, r_8\}\) for these conditions.

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