
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
Given a (nondegenrate) triangle  $ABC$  with positive integer angles (in degrees), construct squares  $BCD_1D_2, ACE_1E_2$  outside the triangle. Given that  $D_1, D_2, E_1, E_2$  all lie on a circle, how many ordered triples  $(\angle A, \angle B, \angle C)$  are possible?

*Proposed by Yang Liu*

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