
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
Given a polynomial 
\[ a_8 x^8 + a_7 x^7 + a_6 x^6 + a_5 x^5 + a_4 x^4 + a_3 x^3 + a_2 x^2 + a_1 x + a_0 = 0 \]
where \( a_0, a_8 \neq 0 \) and all coefficients are integers. Its roots \( r_1, r_2, \ldots, r_8 \) are integers. Additionally, the same roots \( r_1, r_2, \ldots, r_8 \) also satisfy the polynomial
\[ a_0 x^8 + a_1 x^7 + a_2 x^6 + a_3 x^5 + a_4 x^4 + a_5 x^3 + a_6 x^2 + a_7 x + a_8 = 0. \]
Find the number of possible multisets \( S = \{r_1, r_2, \ldots, r_8\} \).

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