
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
Three points \(A, B\), and \(C\) on a sphere of radius \(R\) are connected pairwise by the shorter arcs of great circles. Through the midpoints of the arcs \(AB\) and \(AC\), another great circle is drawn, intersecting the extension of the arc \(BC\) at point \(K\). Find the length of the arc \(CK\) if the length of the arc \(BC\) is \( l (l < \pi R)\). (A great circle is defined as the intersection of the sphere with a plane passing through the center.)

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