
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 $p(x)$ and $q(x)$ be two monic cubic polynomials, and let $s$ be a real number. Two of the roots of $p(x)$ are $s + 2$ and $s + 6$. Two of the roots of $q(x)$ are $s + 4$ and $s + 8$, and
\[p(x) - q(x) = 2s\]
for all real numbers $x$. Find $s$.

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