
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
Given a quadratic polynomial $q(x)$ with real coefficients and leading coefficient 1, for which the equation $q(q(x)) = 1$ is satisfied by exactly three distinct real roots, find $\hat{q}(2)$ where $\hat{q}$ is the unique polynomial with the minimum sum of roots.

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