
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
\[ b_8 x^8 + b_7 x^7 + b_6 x^6 + \dots + b_1 x + b_0 = 0 \]
where all coefficients \( b_i \) are integers, and its roots \( s_1, s_2, \dots, s_8 \) are integers. Additionally, the roots of the polynomial created by reversing these coefficients,
\[ b_0 x^8 + b_1 x^7 + b_2 x^6 + \dots + b_6 x^2 + b_7 x + b_8 = 0 \]
are also \( s_1, s_2, \dots, s_8 \), and the sum of all roots equals to 0. Determine how many possible multisets \( T = \{s_1, s_2, \dots, s_8\} \) exist.

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