
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
Two circles  $K_{1}$  and  $K_{2}$ , centered at  $O_{1}$  and  $O_{2}$  with radii  $1$  and  $\sqrt{2}$  respectively, intersect at  $A$  and  $B$ . Let  $C$  be a point on  $K_{2}$  such that the midpoint of  $AC$  lies on  $K_{1}$ . Find the length of the segment  $AC$  if  $O_{1}O_{2}=2$ 

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