
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 $(0, 0)$, $(2, 1)$, and $(k, 0)$, where $k$ is a positive real number. A line passing through the origin with a slope of $m$ divides the triangle into two regions of equal area. Determine all possible values of $m$ that satisfy this condition.
A) $\frac{1}{3}$  
B) $\frac{1}{4}$  
C) $\frac{1}{5}$  
D) $\frac{1}{2}$

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