
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
Let the coordinates of point \( M \) on the plane be \( (p \times 1994, 7p \times 1994) \), where \( p \) is a prime number. Determine the number of right triangles satisfying the following conditions:
1. All three vertices of the triangle are integer points, and \( M \) is the right-angled vertex;
2. The incenter of the triangle is the origin of the coordinates.

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