[Theorem of tangent and secant; the product of the entire secant and its external part] [Law of Cosines] [Law of Sines]

In triangle $K L M$, it is known that $\angle L K M=\beta, \angle L M K=\gamma, K M=a$. On side $K L$, a point $N$ is taken such that $K N = 2 N L$. A circle is drawn through points $L$ and $N$, tangent to side $K M$ or its extension beyond point $M$.

Find the radius of the circle.
Answer

$\frac{a\left(\frac{5}{3}-2 \sqrt{\frac{2}{3}} \cos \beta\right) \sin \gamma}{2 \sin \beta \sin (\beta+\gamma)}$.