
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 an acute-angled scalene triangle \( ABC \). Outside of this triangle, construct isosceles obtuse triangles \( AB_1C \) and \( BA_1C \) with equal angles \(\alpha\) at their bases \( AC \) and \( BC \). The perpendicular from vertex \( C \) to segment \( A_1B_1 \) intersects the perpendicular bisector of side \( AB \) at point \( C_1 \). Find the angle \( AC_1B \).

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