
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
\[ Q(2) = Q(4) = Q(6) = Q(8) = b \]
and
\[ Q(1) = Q(3) = Q(5) = Q(7) = -b. \]
What is the smallest possible value of \( b \)?

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