A sequence of integers is said to Zigzag if adjacent elements alternate between strictly increasing and strictly decreasing. Note that the sequence may start by either increasing or decreasing. Given a sequence of integers, determine the length of the longest subsequence that Zigzags. For example, consider this sequence:

1 2 3 4 2

There are several Zigzagging subsequences of length $3$:

1 3 2        1 4 2        2 3 2        2 4 2        3 4 2

But there are none of length greater than $3$, so the answer is $3$.

-----Input-----
Each input will consist of a single test case. Note that your program may be run multiple times on different inputs. The first line of input contains an integer $n$ ($1 \le n \le 1000000$) which is the number of integers in the list. Each of the following $n$ lines will have an integer $k$ ($1 \le k \le 1000000$).

-----Output-----
Output a single integer, which is the length of the longest Zigzagging subsequence of the input list.

-----Examples-----
Sample Input:
5
1
2
3
4
2
Sample Output:
3

Source: kattis | Difficulty: MEDIUM


Solutions: