
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
Find the number of right triangles on a plane that satisfy the following conditions:
(1) The three vertices of the triangle are integer points, with the origin as the right-angle vertex;
(2) The coordinates of the incenter \( M \) of the triangle are \( (96 p, 672 p) \), where \( p \) is a prime number.

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