Problem Statement

We have a 2 \times N grid. We will denote the square at the i-th row and j-th column (1 \leq i \leq 2, 1 \leq j \leq N) as (i, j).

You are initially in the top-left square, (1, 1). You will travel to the bottom-right square, (2, N), by repeatedly moving right or down.

The square (i, j) contains A_{i, j} candies. You will collect all the candies you visit during the travel. The top-left and bottom-right squares also contain candies, and you will also collect them.

At most how many candies can you collect when you choose the best way to travel?

Constraints

• 1 \leq N \leq 100
• 1 \leq A_{i, j} \leq 100 (1 \leq i \leq 2, 1 \leq j \leq N)

Sample Input 1

5
3 2 2 4 1
1 2 2 2 1

Sample Output 1

14

The number of collected candies will be maximized when you:

• move right three times, then move down once, then move right once.

Sample Input 2

4
1 1 1 1
1 1 1 1

Sample Output 2

5

You will always collect the same number of candies, regardless of how you travel.

Sample Input 3

7
3 3 4 5 4 5 3
5 3 4 4 2 3 2

Sample Output 3

29

Sample Input 4

1
2
3

Sample Output 4

5

Problem Statement

There are N people standing on the x-axis. Let the coordinate of Person i be x_i. For every i, x_i is an integer between 0 and 10^9 (inclusive). It is possible that more than one person is standing at the same coordinate.

You will given M pieces of information regarding the positions of these people. The i-th piece of information has the form (L_i, R_i, D_i). This means that Person R_i is to the right of Person L_i by D_i units of distance, that is, x_{R_i} - x_{L_i} = D_i holds.

It turns out that some of these M pieces of information may be incorrect. Determine if there exists a set of values (x_1, x_2, ..., x_N) that is consistent with the given pieces of information.

Constraints

• 1 \leq N \leq 100 000
• 0 \leq M \leq 200 000
• 1 \leq L_i, R_i \leq N (1 \leq i \leq M)
• 0 \leq D_i \leq 10 000 (1 \leq i \leq M)
• L_i \neq R_i (1 \leq i \leq M)
• If i \neq j, then (L_i, R_i) \neq (L_j, R_j) and (L_i, R_i) \neq (R_j, L_j).
• D_i are integers.

Sample Input 1

3 3
1 2 1
2 3 1
1 3 2

Sample Output 1

Yes

Some possible sets of values (x_1, x_2, x_3) are (0, 1, 2) and (101, 102, 103).

Sample Input 2

3 3
1 2 1
2 3 1
1 3 5

Sample Output 2

No

If the first two pieces of information are correct, x_3 - x_1 = 2 holds, which is contradictory to the last piece of information.

Sample Input 3

4 3
2 1 1
2 3 5
3 4 2

Sample Output 3

Yes

Sample Input 4

10 3
8 7 100
7 9 100
9 8 100

Sample Output 4

No

Sample Input 5

100 0

Sample Output 5

Yes

Problem Statement

We have a graph with N vertices and M edges, and there are two people on the graph: Takahashi and Aoki.

The i-th edge connects Vertex U_i and Vertex V_i. The time it takes to traverse this edge is D_i minutes, regardless of direction and who traverses the edge (Takahashi or Aoki).

Takahashi departs Vertex S and Aoki departs Vertex T at the same time. Takahashi travels to Vertex T and Aoki travels to Vertex S, both in the shortest time possible. Find the number of the pairs of ways for Takahashi and Aoki to choose their shortest paths such that they never meet (at a vertex or on an edge) during the travel, modulo 10^9 + 7.

Constraints

• 1 \leq N \leq 100 000
• 1 \leq M \leq 200 000
• 1 \leq S, T \leq N
• S \neq T
• 1 \leq U_i, V_i \leq N (1 \leq i \leq M)
• 1 \leq D_i \leq 10^9 (1 \leq i \leq M)
• If i \neq j, then (U_i, V_i) \neq (U_j, V_j) and (U_i, V_i) \neq (V_j, U_j).
• U_i \neq V_i (1 \leq i \leq M)
• D_i are integers.
• The given graph is connected.

Sample Input 1

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

Sample Output 1

2

There are two ways to choose shortest paths that satisfies the condition:

• Takahashi chooses the path 1 \rightarrow 2 \rightarrow 3, and Aoki chooses the path 3 \rightarrow 4 \rightarrow 1.
• Takahashi chooses the path 1 \rightarrow 4 \rightarrow 3, and Aoki chooses the path 3 \rightarrow 2 \rightarrow 1.

Sample Input 2

3 3
1 3
1 2 1
2 3 1
3 1 2

Sample Output 2

2

Sample Input 3

3 3
1 3
1 2 1
2 3 1
3 1 2

Sample Output 3

2

Sample Input 4

8 13
4 2
7 3 9
6 2 3
1 6 4
7 6 9
3 8 9
1 2 2
2 8 12
8 6 9
2 5 5
4 2 18
5 3 7
5 1 515371567
4 8 6

Sample Output 4

6

Problem Statement

For a positive integer n, let us define f(n) as the number of digits in base 10.

You are given an integer S. Count the number of the pairs of positive integers (l, r) (l \leq r) such that f(l) + f(l + 1) + ... + f(r) = S, and find the count modulo 10^9 + 7.

Constraints

• 1 \leq S \leq 10^8

Sample Input 1

1

Sample Output 1

9

There are nine pairs (l, r) that satisfies the condition: (1, 1), (2, 2), ..., (9, 9).

Sample Input 2

2

Sample Output 2

98

There are 98 pairs (l, r) that satisfies the condition, such as (1, 2) and (33, 33).

Sample Input 3

123

Sample Output 3

460191684

Sample Input 4

36018

Sample Output 4

966522825

Sample Input 5

1000

Sample Output 5

184984484