
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
Let $b > 0$, and let $Q(x)$ be a polynomial with integer coefficients such that
\[Q(1) = Q(4) = Q(7) = Q(10) = b\]
and
\[Q(2) = Q(5) = Q(8) = Q(11) = -b.\]
What is the smallest possible value of $b$?

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