
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
Consider a triangle with vertices at \((0,0)\), \((2,2)\), and \((8k,0)\), where \(k\) is a positive real number. A line \(y = kx\) divides this triangle into two regions of equal area. What is the sum of all possible values of \(k\)?
\(\textbf{A}\) \(-\frac{1}{4}\) \quad \textbf{B}\ \(-\frac{1}{8}\) \quad \textbf{C}\ \(\frac{1}{4}\) \quad \textbf{D}\ \(\frac{1}{8}\)

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