Description

Problem D
Wavio Sequence
Input: Standard Input

Output: Standard Output

Time Limit: 2 Seconds

Wavio is a sequence of integers. It has some interesting properties.

? Wavio is of odd length i.e. L = 2*n + 1.

? The first (n+1) integers of Wavio sequence makes a strictly increasing sequence.

? The last (n+1) integers of Wavio sequence makes a strictly decreasing sequence.

? No two adjacent integers are same in a Wavio sequence.

For example 1, 2, 3, 4, 5, 4, 3, 2, 0 is an Wavio sequence of length 9. But 1, 2, 3, 4, 5, 4, 3, 2, 2 is not a valid wavio sequence. In this problem, you will be given a sequence of integers. You have to find out the length of the longest Wavio sequence which is a subsequence of the given sequence. Consider, the given sequence as :

1 2 3 2 1 2 3 4 3 2 1 5 4 1 2 3 2 2 1.

Here the longest Wavio sequence is : 1 2 3 4 5 4 3 2 1. So, the output will be 9.

Input

The input file contains less than 75 test cases. The description of each test case is given below: Input is terminated by end of file.

Each set starts with a postive integer, N(1<=N<=10000). In next few lines there will be N integers.

Output

For each set of input print the length of longest wavio sequence in a line.

Sample Input Output for Sample Input

10 

1 2 3 4 5 4 3 2 1 10 

19 

1 2 3 2 1 2 3 4 3 2 1 5 4 1 2 3 2 2 1 

5 

1 2 3 4 5

9 

9 

1

Problemsetter: Md. Kamruzzaman, Member of Elite Problemsetters' Panel

Wavio是一个整数序列，具有以下特性：
1、Wavio序列的长度是奇数， 即 L = 2 * n + 1;
2、Wavio序列前 n+1 个整数是递增序列
3、Wavio序列后 n+1 个整数是递减序列
如示例 1 2 3 4 5 4 3 2 1 10
最长的 Wavio序列 为 1 2 3 4 5 4 3 2 1 ，所以答案为9
对于输入序列中的一个整数 ai ，我们设以 ai 为尾的前缀的最长递增序列的长度为Fi ，如在示例1中对已第3个整数3，从头开始，以3为尾的递增序列为1 2 3 ，所以F3=3；
以ai为首的后缀的递减序列的长度为Gi, 如示例1中第3个整数，以3为开始，递减序列为3 2 1，所以G3=3 (可以看做求从最后一个元素出发，到3这个位置的最大递增序列，为1 2 3，所以G3=3)
在我们找递减序列的时候，可以看做从最后一个元素出发，到当前位置的最大递增序列，这样我们对于每一个元素ai 我们可以先求出 a0到ai最大递增序列 和 an-1到ai的最大递增序列 ，这样我们可以得到ai的Wavio序列的值为
2*min( Fi , Gi) - 1 ,最后的结果是这些Wavio序列中的一个最大值。

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