
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  is called a  $\text{lattice point}$  if both of its coordinates are integers. Let  $A$   be the point  $(12,84)$ . Find the number of right  angled triangles  $ABC$   in the coordinate plane  $B$  and  $C$  are lattice points, having a right angle at vertex  $A$   and whose incenter is at the origin  $(0,0)$ .

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