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

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