
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
Suppose  $X, Y, Z$  are collinear points in that order such that  $XY = 1$  and  $YZ = 3$ . Let  $W$  be a point such that  $YW = 5$ , and define  $O_1$  and  $O_2$  as the circumcenters of triangles  $\triangle WXY$  and  $\triangle WYZ$ , respectively. What is the minimum possible length of segment  $\overline{O_1O_2}$ ?

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