
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
Circle  $\omega_1$  with center  $K$  of radius  $4$  and circle  $\omega_2$  of radius  $6$  intersect at points  $W$  and  $U$ . If the incenter of  $\triangle KWU$  lies on circle  $\omega_2$ , find the length of  $\overline{WU}$ . (Note: The incenter of a triangle is the intersection of the angle bisectors of the angles of the triangle)

*Proposed by Bradley Guo*

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