
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
A triangle \( \triangle ABC \) with an angle \( \angle ABC = 135^\circ \) is inscribed in a circle \( \omega \). The lines tangent to \( \omega \) at \( A \) and \( C \) intersect at point \( D \). Find \( \angle ABD \), given that \( AB \) bisects the segment \( CD \). Answer: \( 90^\circ \).

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