
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
\[ x^6 + b_5 x^5 + b_4 x^4 + b_3 x^3 + b_2 x^2 + b_1 x + b_0, \]
where \( b_i \in \{0, 1, 2\} \) for all \(0 \leq i \leq 5\). Find the number of such polynomials that have exactly one integer root equal to 2.

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