
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 $a > 0$, and let $P(x)$ be a polynomial with integer coefficients such that
\[ P(1) = P(3) = P(5) = P(7) = a \]
and
\[ P(2) = P(4) = P(6) = P(8) = P(10) = -a. \]
Determine the smallest possible value of $a$.

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