
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
Consider a circle of radius  $4$  with center  $O_1$ , a circle of radius  $2$  with center  $O_2$  that lies on the circumference of circle  $O_1$ , and a circle of radius  $1$  with center  $O_3$  that lies on the circumference of circle  $O_2$ . The centers of the circle are collinear in the order  $O_1$ ,  $O_2$ ,  $O_3$ . Let  $A$  be a point of intersection of circles  $O_1$  and  $O_2$  and  $B$  be a point of intersection of circles  $O_2$  and  $O_3$  such that  $A$  and  $B$  lie on the same semicircle of  $O_2$ . Compute the length of  $AB$ .

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