
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  $ABCD$  be a nondegenerate isosceles trapezoid with integer side lengths such that  $BC \parallel AD$  and  $AB=BC=CD$ . Given that the distance between the incenters of triangles  $ABD$  and  $ACD$  is  $8!$ , determine the number of possible lengths of segment  $AD$ .

*Ray Li*

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