
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
\[x^7 + b_6 x^6 + b_5 x^5 + \dots + b_1 x + b_0,\]
where \(b_i \in \{0,1\}\) for all \(0 \le i \le 6\). Find the number of such polynomials that have exactly three different integer roots, one of which must be zero.

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