
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
A triangle has vertices at \( (0, 0) \), \( (2, 2) \), and \( (8k, 0) \). A line \( y = 2kx \) divides the triangle into two triangles of equal area. Find the sum of all possible values of \( k \).
**A)** \( -\frac{1}{2} \)  
**B)** \( -\frac{1}{4} \)  
**C)** \( \frac{1}{4} \)  
**D)** \( \frac{1}{2} \)

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