Design an $O(nlogn)$ divide$-$and$-$conquer algorithm for the closest pair task.

Input: A set of points in the plane, $\{p_1=(x_1,y_1),p_2=(x_2,y_2),$dots,$p_n=(x_n,y_n)\}$

$2$

Output: The closest pair of points: that is$,$ the pair $p_i{p}_j$ for which the distance between $p_i$ and $p_j,$ that is$,$

$\sqrt[\phantom{2}]{(x_i-x_j{)}^2+(y_i-y_j{)}^2}$

is minimized.