
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
It is required to construct a right triangle in the coordinate plane so that its hypotenuse is parallel to the line $y = mx + b$ and its legs are along the $x$ and $y$ axes respectively. Determine the number of different constants $m$ for which the medians to the hypotenuse and one of the legs can simultaneously fall on the lines $y = 2x + 1$ and $y = 5x + 2$.

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