
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 square of side length  $1$ , and let  $P$  be a variable point on  $\overline{CD}$ .  Denote by  $Q$  the intersection point of the angle bisector of  $\angle APB$  with  $\overline{AB}$ .  The set of possible locations for  $Q$  as  $P$  varies along  $\overline{CD}$  is a line segment; what is the length of this segment?

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