
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 the plane that meet the following conditions:
(1) The three vertices of the triangle are all integer points, with the origin as the right angle vertex.
(2) The coordinates of the incenter $M$ of the triangle are $(96p, 672p)$, where $p$ is a prime number.

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