
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 legs parallel to the x and y axes such that the median to the midpoint of the leg parallel to the x-axis lies on the line y = 3x + 1 and the median to the midpoint of the leg parallel to the y-axis lies on the line y = (2m+1)x + 3. Determine the number of possible values for the constant m such that the triangle exists.

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