
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
Let  $ABC$  be a triangle, not right-angled, with positive integer angle measures (in degrees) and circumcenter  $O$ . Say that a triangle  $ABC$  is *good* if the following three conditions hold:

(a) There exists a point  $P\neq A$  on side  $AB$  such that the circumcircle of  $\triangle POA$  is tangent to  $BO$ .
(b) There exists a point  $Q\neq A$  on side  $AC$  such that the circumcircle of  $\triangle QOA$  is tangent to  $CO$ .
(c) The perimeter of  $\triangle APQ$  is at least  $AB+AC$ .

Determine the number of ordered triples  $(\angle A, \angle B,\angle C)$  for which  $\triangle ABC$  is good.

*Proposed by Vincent Huang*

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