
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
Find the smallest positive integer \( b \) for which there exists a polynomial \( Q(x) \) with integer coefficients such that:
\[ Q(1) = Q(4) = Q(7) = Q(10) = b, \]
\[ Q(2) = Q(5) = Q(8) = Q(11) = -2b. \]
\[ Q(3) = Q(6) = Q(9) = Q(12) = 3b. \]

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