
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 all polynomials of the form
\[x^{11} + b_{10}x^{10} + b_9x^9 + \dots + b_1x + b_0,\]
where $b_i \in \{0,1\}$ for all $0 \leq i \leq 10$. Find the number of such polynomials that have exactly three different integer roots.

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