
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
\[x^{10} + a_9 x^9 + a_8 x^8 + \dots + a_1 x + a_0,\]
where \(a_i \in \{0,1\}\) for all \(0 \le i \le 9.\) Find the number of such polynomials that have exactly three different integer roots, specifically 0, 1, and -1.

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