
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
Determine the number of possible integer multisets $S = \{r_1, r_2, \dots, r_6\}$ for which each $r_i$ is a root of the polynomial
\[ b_6 x^6 + b_5 x^5 + b_4 x^4 + b_3 x^3 + b_2 x^2 + b_1 x + b_0 = 0,\]
where all $b_i$ are integers, $b_6 \neq 0$, $b_0 \neq 0$, and the roots of the polynomial
\[b_0 x^6 + b_1 x^5 + b_2 x^4 + b_3 x^3 + b_4 x^2 + b_5 x + b_6 = 0\]
are also exactly $r_1, r_2, \dots, r_6$.

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