
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
Let  $AB = 10$  be a diameter of circle  $P$ . Pick point  $C$  on the circle such that  $AC = 8$ . Let the circle with center  $O$  be the incircle of  $\vartriangle ABC$ . Extend line  $AO$  to intersect circle  $P$  again at  $D$ . Find the length of  $BD$ .

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