
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
Determine the number of different constants $m$ for which it is possible to construct a right triangle in the coordinate plane with legs parallel to the $x$ and $y$ axes, such that the medians to the midpoints of the legs lie on the lines $y = 2x + 1$ and $y = mx + 3$.

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