
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
X is the point (1994p, 7·1994p), where p is a prime, and O is the point (0, 0). How many triangles XYZ have Y and Z at lattice points, incenter at O, and YXZ as a right-angle?

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