Polycarp is a frequent user of the very popular messenger. He's chatting with his friends all the time. He has 𝑛 friends, numbered from 1 to 𝑛.

Recall that a permutation of size 𝑛 is an array of size 𝑛 such that each integer from 1 to 𝑛 occurs exactly once in this array.

You are given 𝑛 arrays 𝑎1, 𝑎2, ..., 𝑎𝑛; each array consists of exactly 𝑚 integers. We denote the 𝑦-th element of the 𝑥-th array as 𝑎𝑥,𝑦.

You have to choose two arrays 𝑎𝑖 and 𝑎𝑗 (1≤𝑖,𝑗≤𝑛, it is possible that 𝑖=𝑗). After that, you will obtain a new array 𝑏 consisting of 𝑚 integers, such that for every $k \in [1, m]$ $b_k = \max(a_{i, k}, a_{j, k})$ .

Your goal is to choose $𝑖$ and $𝑗$ so that the value of min$\min \limits_{k = 1}^{m} b_k$ is maximum possible.

Filled with optimism, Hyunuk will host a conference about how great this new year will be!

The conference will have 𝑛 lectures. Hyunuk has two candidate venues 𝑎 and 𝑏. For each of the 𝑛 lectures, the speaker specified two time intervals [𝑠𝑎𝑖,𝑒𝑎𝑖] (𝑠𝑎𝑖≤𝑒𝑎𝑖) and [𝑠𝑏𝑖,𝑒𝑏𝑖] (𝑠𝑏𝑖≤𝑒𝑏𝑖). If the conference is situated in venue 𝑎, the lecture will be held from 𝑠𝑎𝑖 to 𝑒𝑎𝑖, and if the conference is situated in venue 𝑏, the lecture will be held from 𝑠𝑏𝑖 to 𝑒𝑏𝑖. Hyunuk will choose one of these venues and all lectures will be held at that venue.

Two lectures are said to overlap if they share any point in time in common. Formally, a lecture held in interval [𝑥,𝑦] overlaps with a lecture held in interval [𝑢,𝑣] if and only if max(𝑥,𝑢)≤min(𝑦,𝑣).

You are given a sequence 𝑎1,𝑎2,…,𝑎𝑛, consisting of integers.

You can apply the following operation to this sequence: choose some integer 𝑥 and move all elements equal to 𝑥 either to the beginning, or to the end of 𝑎. Note that you have to move all these elements in one direction in one operation.

The great rout will be held this evening in the palace of his majesty Nassah II, the king of Occorom. There are n guests invited. While they are preparing evening dresses and collecting fresh rumors to talk about, the chief valet of the palace has a tricky task to solve: choose the right order for persons to arrive to the palace.

Alice is playing a game with her good friend, Marisa.

There are 𝑛 boxes arranged in a line, numbered with integers from 1 to 𝑛 from left to right. Marisa will hide a doll in one of the boxes. Then Alice will have 𝑚 chances to guess where the doll is. If Alice will correctly guess the number of box, where doll is now, she will win the game, otherwise, her friend will win the game.

outputstandard output
This is a harder version of the problem. In this version, 𝑛≤300000.

Vasya is an experienced developer of programming competitions' problems. As all great minds at some time, Vasya faced a creative crisis. To improve the situation, Petya gifted him a string consisting of opening and closing brackets only. Petya believes, that the beauty of the bracket string is a number of its cyclical shifts, which form a correct bracket sequence.