
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 legs are parallel to the \( x \) and \( y \) axes, and the medians to the midpoints of the legs lie on the lines \( y = 4x + 2 \) and \( y = nx + 3 \). Determine the number of different constants \( n \) for which such a triangle exists.

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