
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
In a convex quadrilateral \(ABCD\), the centers of the circumscribed and inscribed circles of triangle \(ABC\) coincide with the centers of the inscribed and circumscribed circles of triangle \(ADC\), respectively. It is known that \(AB = 1\). Find the lengths of the other sides and the angles of the quadrilateral.

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