
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
In the coordinate plane, a point with both integer x-coordinate and y-coordinate is called a lattice point, and a triangle with all vertices as lattice points is called a lattice triangle. How many lattice right triangles \( \mathrm{Rt} \triangle OAB \) exist with the incenter at \( I(2015, 7 \times 2015) \) and the right-angle vertex at the origin \( O \)?

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