Mountain

Figure 1 A mountain with four peaks

See Figure 1. What do you see? Yes, it is a mountain. Let’s call such a figure a mountain. To be more precise, a mountain is an $x$-monotone polygonal chain consisting of at least two line segments whose slopes alternate between $+1$ and $−1$; the leftmost segment is a half-line (ray) of slope $+1$, and the rightmost segment is a half-line of slope $−1$. Every mountain has one or more peaks, which are endpoints incident to a segment of slope $+1$ to the left and a segment of slope $−1$ to the right; in other words, a peak is a locally highest point. In Figure 1, you can see a mountain consisting of eight segments, and it has four peaks, marked by small dots.

In this problem, you are given a set $P$ of $n$ points in the plane as input, and your task is to find a mountain with the maximum number of peaks that are chosen from $P$.

Figure 2 Two mountains with four and seven peaks, respectively, chosen from a given set $P$ of $15$ points

An example is illustrated in Figure 2, in which the input set $P$ of $n = 15$ points, marked by small circles, is given, and to the left you can see a mountain with four peaks that belong to $P$, while to the right there is a mountain with seven peaks that belong to $P$. Since there is no such mountain with eight peaks chosen from $P$, the one to the right is a correct answer for the problem. See Sample Input/Output 2 below

Your program is to read from standard input. The input starts with a line containing a single integer $n$ ($1 ≤ n ≤ 500\\,000$), where $n$ is the number of points in the input set $P$ of points in the plane. In each of the following $n$ lines, given are two integers $x$ and $y$, both ranging from $−10^6$ to $10^6$, inclusively, that represent the $x$- and $y$-coordinates of an input point $(x, y)$ in $P$. You may assume that no two input points have the same coordinates.

Your program is to write to standard output. Print exactly one line. The line should contain an integer that represents the number of peaks of a mountain with the maximum number of peaks that are chosen from $P$.