
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
Cyclic quadrilateral  $ABCD$  satisfies  $\angle ADC = 2 \cdot \angle BAD = 80^\circ$  and  $\overline{BC} = \overline{CD}$ . Let the angle bisector of  $\angle BCD$  meet  $AD$  at  $P$ . What is the measure, in degrees, of  $\angle BP D$ ?


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