
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 \( M \) be the point with coordinates \((p \cdot 2002, 7p \cdot 2002)\) in the coordinate plane, where \( p \) is a rational number. Find the number of right-angled triangles that satisfy the following conditions:
1. All three vertices of the triangle are lattice points, and \( M \) is the right-angled vertex.
2. The incenter of the triangle is the origin.

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