
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 scalene triangle $ABC$, the angle bisector $AD$ intersects the circumcircle of the triangle at point $P$. Point $I$ is the incenter of the triangle $ABC$. It turns out that $ID = DP$. Find the ratio $AI: ID$. (20 points)

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