
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 triangle \(ABC\), the angle bisector \(AD\) is drawn. It is known that the centers of the incircle of triangle \(ABD\) and the circumcircle of triangle \(ABC\) coincide. Find \(CD\) if \(AC = \sqrt{5} + 1\). The answer should not include the designations of trigonometric functions and their inverses.

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