
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
On the sides \( AB \) and \( BC \) of triangle \( ABC \), right triangles \( APB \) and \( BQC \) with identical angles \(\varphi\) at their common vertex \( B \) are constructed externally on these sides as hypotenuses. Find the angles of triangle \( PQK \), where \( K \) is the midpoint of side \( AC \).

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