
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
Points \( A_{1}, B_{1}, C_{1} \) are the intersections of the extensions of the altitudes of the acute-angled triangle \( A B C \) with the circumcircle of \( A B C \). The incircle of the triangle \( A_{1} B_{1} C_{1} \) touches one of the sides of \( A B C \), and one of the angles of the triangle \( A B C \) is \( 50^{\circ} \). Find the other two angles of the triangle \( A B C \).

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