
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
Construct a right triangle in the coordinate plane with vertices at $A(0, 0)$, $B(0, 3k)$, and $C(4k, 0)$ (where $k > 0$) such that this triangle's medians to the midpoints of the legs lie on the lines $y = 3x + k$ and $y = mx + k$. Find the number of different constants $m$ for which such a triangle exists.

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