Closest Pair

You are given two finite sets of points in the plane, $P$ and $Q$. A closest pair of $P$ and $Q$ is a pair $(p, q)$ with $p \in P$ and $q \in Q$ whose distance is the smallest among all pairs $(p', q')$ with $p' \in P$ and $q' \in Q$.

In this problem the distance between two points $a$ and $b$ is defined as follows.

$d(a, b) = |x_a - x_b| + |y_a - y_b|$

Here $x_a$ and $y_a$ are the $x$ and $y$ coordinates of point $a$, and $x_b$ and $y_b$ are the $x$ and $y$ coordinates of point $b$. So a pair $(p, q)$ with $p \in P$ and $q \in Q$ is a closest pair of $P$ and $Q$ if and only if

$d(p, q) = \min \{ d(p', q') : p' \in P,\ q' \in Q \}$

Given $P$ and $Q$, write a program that computes the distance of a closest pair and the number of distinct closest pairs. Two pairs count as distinct when the point taken from $P$ differs or the point taken from $Q$ differs.

The input points satisfy these two properties.

1. There are integers $c_1$ and $c_2$ such that every point of $P$ lies on the line $y = c_1$ and every point of $Q$ lies on the line $y = c_2$.
2. No two points of $P$ have the same coordinates, and no two points of $Q$ have the same coordinates.

Your program reads from standard input. The input consists of four lines.

The first line contains two integers $n$ and $m$ ($1 \le n \le 5 \times 10^5$, $1 \le m \le 5 \times 10^5$), where $n$ is the number of points in $P$ and $m$ is the number of points in $Q$.

The second line contains two integers $c_1$ and $c_2$ ($-10^8 \le c_1, c_2 \le 10^8$) in this order, separated by a single space.

The third line contains $n$ distinct integers between $-10^8$ and $10^8$, separated by single spaces. They are the $x$ coordinates of the points in $P$, whose $y$ coordinates all equal $c_1$.

The fourth line contains $m$ distinct integers between $-10^8$ and $10^8$, separated by single spaces. They are the $x$ coordinates of the points in $Q$, whose $y$ coordinates all equal $c_2$.

Your program writes to standard output. Print exactly one line holding two integers separated by a single space: the distance of a closest pair of $P$ and $Q$, then the number of closest pairs of $P$ and $Q$.