131. Three points $A, B$, and $C$, located on the surface of a sphere with radius $R$, are connected by arcs of great circles, each less than a semicircle. Another great circle is drawn through the midpoints of the arcs $\overrightarrow{A B}$ and $\widehat{A C}$, intersecting the extension of $\widetilde{B} C$ at point $K$. Find the length of the arc $\widetilde{C K}$, if $|\overrightarrow{B C}|=l(l<\pi R)$.
131. Chord $B C$ is parallel to any plane passing through the midpoints of chords $A B$ and $A C$. Therefore, chord $B C$ is parallel to the plane passing through the center of the sphere and the midpoints of arcs $\overrightarrow{A B}$ and $\overrightarrow{A C}$. From this, it follows that the great circle passing through $B$ and $C$, and the great circle passing through the midpoints of arcs $\widetilde{A B}$ and $\overline{A C}$, intersect at two points $K$ and $K_{1}$ in such a way that the diameter $K K_{1}$ is parallel to chord $B C$.

Answer: $\frac{\pi R}{2} \pm \frac{l}{2}$.