Problem Statement

There are N candles placed on a number line. The i-th candle from the left is placed on coordinate x_i. Here, x_1 < x_2 < ... < x_N holds.

Initially, no candles are burning. Snuke decides to light K of the N candles.

Now, he is at coordinate 0. He can move left and right along the line with speed 1. He can also light a candle when he is at the same position as the candle, in negligible time.

Find the minimum time required to light K candles.

Constraints

• 1 \leq N \leq 10^5
• 1 \leq K \leq N
• x_i is an integer.
• |x_i| \leq 10^8
• x_1 < x_2 < ... < x_N

Sample Input 1

5 3
-30 -10 10 20 50

Sample Output 1

40

He should move and light candles as follows:

• Move from coordinate 0 to -10.
• Light the second candle from the left.
• Move from coordinate -10 to 10.
• Light the third candle from the left.
• Move from coordinate 10 to 20.
• Light the fourth candle from the left.

Sample Input 2

3 2
10 20 30

Sample Output 2

20

Sample Input 3

1 1
0

Sample Output 3

0

• There may be a candle placed at coordinate 0.

Sample Input 4

8 5
-9 -7 -4 -3 1 2 3 4

Sample Output 4

10

Problem Statement

We will define the median of a sequence b of length M, as follows:

• Let b' be the sequence obtained by sorting b in non-decreasing order. Then, the value of the (M / 2 + 1)-th element of b' is the median of b. Here, / is integer division, rounding down.

For example, the median of (10, 30, 20) is 20; the median of (10, 30, 20, 40) is 30; the median of (10, 10, 10, 20, 30) is 10.

Snuke comes up with the following problem.

You are given a sequence a of length N. For each pair (l, r) (1 \leq l \leq r \leq N), let m_{l, r} be the median of the contiguous subsequence (a_l, a_{l + 1}, ..., a_r) of a. We will list m_{l, r} for all pairs (l, r) to create a new sequence m. Find the median of m.

Constraints

• 1 \leq N \leq 10^5
• a_i is an integer.
• 1 \leq a_i \leq 10^9

Sample Input 1

3
10 30 20

Sample Output 1

30

The median of each contiguous subsequence of a is as follows:

• The median of (10) is 10.
• The median of (30) is 30.
• The median of (20) is 20.
• The median of (10, 30) is 30.
• The median of (30, 20) is 30.
• The median of (10, 30, 20) is 20.

Thus, m = (10, 30, 20, 30, 30, 20) and the median of m is 30.

Sample Input 2

1
10

Sample Output 2

10

Sample Input 3

10
5 9 5 9 8 9 3 5 4 3

Sample Output 3

8

Problem Statement

Let N be an even number.

There is a tree with N vertices. The vertices are numbered 1, 2, ..., N. For each i (1 \leq i \leq N - 1), the i-th edge connects Vertex x_i and y_i.

Snuke would like to decorate the tree with ribbons, as follows.

First, he will divide the N vertices into N / 2 pairs. Here, each vertex must belong to exactly one pair. Then, for each pair (u, v), put a ribbon through all the edges contained in the shortest path between u and v.

Snuke is trying to divide the vertices into pairs so that the following condition is satisfied: "for every edge, there is at least one ribbon going through it." How many ways are there to divide the vertices into pairs, satisfying this condition? Find the count modulo 10^9 + 7. Here, two ways to divide the vertices into pairs are considered different when there is a pair that is contained in one of the two ways but not in the other.

Constraints

• N is an even number.
• 2 \leq N \leq 5000
• 1 \leq x_i, y_i \leq N
• The given graph is a tree.

Sample Input 1

4
1 2
2 3
3 4

Sample Output 1

2

There are three possible ways to divide the vertices into pairs, as shown below, and two satisfy the condition: the middle one and the right one.

Sample Input 2

4
1 2
1 3
1 4

Sample Output 2

3

There are three possible ways to divide the vertices into pairs, as shown below, and all of them satisfy the condition.

Sample Input 3

6
1 2
1 3
3 4
1 5
5 6

Sample Output 3

10

Sample Input 4

10
8 5
10 8
6 5
1 5
4 8
2 10
3 6
9 2
1 7

Sample Output 4

672

Problem Statement

There are N robots and M exits on a number line. The N + M coordinates of these are all integers and all distinct. For each i (1 \leq i \leq N), the coordinate of the i-th robot from the left is x_i. Also, for each j (1 \leq j \leq M), the coordinate of the j-th exit from the left is y_j.

Snuke can repeatedly perform the following two kinds of operations in any order to move all the robots simultaneously:

• Increment the coordinates of all the robots on the number line by 1.
• Decrement the coordinates of all the robots on the number line by 1.

Each robot will disappear from the number line when its position coincides with that of an exit, going through that exit. Snuke will continue performing operations until all the robots disappear.

When all the robots disappear, how many combinations of exits can be used by the robots? Find the count modulo 10^9 + 7. Here, two combinations of exits are considered different when there is a robot that used different exits in those two combinations.

Constraints

• 1 \leq N, M \leq 10^5
• 1 \leq x_1 < x_2 < ... < x_N \leq 10^9
• 1 \leq y_1 < y_2 < ... < y_M \leq 10^9
• All given coordinates are integers.
• All given coordinates are distinct.

Sample Input 1

2 2
2 3
1 4

Sample Output 1

3

The i-th robot from the left will be called Robot i, and the j-th exit from the left will be called Exit j. There are three possible combinations of exits (the exit used by Robot 1, the exit used by Robot 2) as follows:

• (Exit 1, Exit 1)
• (Exit 1, Exit 2)
• (Exit 2, Exit 2)

Sample Input 2

3 4
2 5 10
1 3 7 13

Sample Output 2

8

The exit for each robot can be chosen independently, so there are 2^3 = 8 possible combinations of exits.

Sample Input 3

4 1
1 2 4 5
3

Sample Output 3

1

Every robot uses Exit 1.

Sample Input 4

4 5
2 5 7 11
1 3 6 9 13

Sample Output 4

6

Sample Input 5

10 10
4 13 15 18 19 20 21 22 25 27
1 5 11 12 14 16 23 26 29 30

Sample Output 5

22