
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 \(C\), located on the surface of a sphere of radius \(R\), are pairwise connected by arcs of great circles, each less than a semicircle. Through the midpoints of the arcs \(\overrightarrow{AB}\) and \(\widehat{AC}\), another great circle is drawn, intersecting the extension of \(\widetilde{BC}\) at point \(K\). Find the length of the arc \(\widetilde{CK}\), given \(|\overrightarrow{BC}| = l\) (with \(l < \pi R\)).

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