Question 58
A triangle $ABC$ has the property that $|AB| + |AC| = 3|BC|$. Let $T$ be the point on line segment $AC$ satisfying $|AC| = 4|AT|$. Let $K$ and $L$ be points on the interior of line segments $AB$ and $AC$, respectively, such that $KL \parallel BC$, and $KL$ is tangent to the incircle of $\triangle ABC$. Let $S$ be the intersection of $BT$ and $KL$. Determine the ratio $\frac{|SL|}{|KL|}$.