
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 one leg along the $x$-axis and the other along the $y$-axis. The medians to the midpoints of these legs must lie on the lines $y = 4x + 3$ and $y = mx + 5$. Determine the number of different values of $m$ such that the triangle's vertices are $(-a, 0)$, $(0, b)$, and $(0, 0)$, where $a, b > 0$.

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