Problem Statement

Iroha is very particular about numbers. There are K digits that she dislikes: D_1, D_2, ..., D_K.

She is shopping, and now paying at the cashier. Her total is N yen (the currency of Japan), thus she has to hand at least N yen to the cashier (and possibly receive the change).

However, as mentioned before, she is very particular about numbers. When she hands money to the cashier, the decimal notation of the amount must not contain any digits that she dislikes. Under this condition, she will hand the minimum amount of money.

Find the amount of money that she will hand to the cashier.

Constraints

• 1 ≦ N < 10000
• 1 ≦ K < 10
• 0 ≦ D_1 < D_2 < … < D_K≦9
• \{D_1,D_2,...,D_K\} ≠ \{1,2,3,4,5,6,7,8,9\}

Sample Input 1

1000 8
1 3 4 5 6 7 8 9

Sample Output 1

2000

She dislikes all digits except 0 and 2.

The smallest integer equal to or greater than N=1000 whose decimal notation contains only 0 and 2, is 2000.

Sample Input 2

9999 1
0

Sample Output 2

9999

Problem Statement

We have a large square grid with H rows and W columns. Iroha is now standing in the top-left cell. She will repeat going right or down to the adjacent cell, until she reaches the bottom-right cell.

However, she cannot enter the cells in the intersection of the bottom A rows and the leftmost B columns. (That is, there are A×B forbidden cells.) There is no restriction on entering the other cells.

Find the number of ways she can travel to the bottom-right cell.

Since this number can be extremely large, print the number modulo 10^9+7.

Constraints

• 1 ≦ H, W ≦ 100,000
• 1 ≦ A < H
• 1 ≦ B < W

Sample Input 1

2 3 1 1

Sample Output 1

2

We have a 2×3 grid, but entering the bottom-left cell is forbidden. The number of ways to travel is two: "Right, Right, Down" and "Right, Down, Right".

Sample Input 2

10 7 3 4

Sample Output 2

3570

There are 12 forbidden cells.

Sample Input 3

100000 100000 99999 99999

Sample Output 3

1

Sample Input 4

100000 100000 44444 55555

Sample Output 4

738162020

Problem Statement

Haiku is a short form of Japanese poetry. A Haiku consists of three phrases with 5, 7 and 5 syllables, in this order.

Iroha is looking for X,Y,Z-Haiku (defined below) in integer sequences.

Consider all integer sequences of length N whose elements are between 1 and 10, inclusive. Out of those 10^N sequences, how many contain an X,Y,Z-Haiku?

Here, an integer sequence a_0, a_1, ..., a_{N-1} is said to contain an X,Y,Z-Haiku if and only if there exist four indices x, y, z, w (0 ≦ x < y < z < w ≦ N) such that all of the following are satisfied:

• a_x + a_{x+1} + ... + a_{y-1} = X
• a_y + a_{y+1} + ... + a_{z-1} = Y
• a_z + a_{z+1} + ... + a_{w-1} = Z

Since the answer can be extremely large, print the number modulo 10^9+7.

Constraints

• 3 ≦ N ≦ 40
• 1 ≦ X ≦ 5
• 1 ≦ Y ≦ 7
• 1 ≦ Z ≦ 5

Sample Input 1

3 5 7 5

Sample Output 1

1

Here, the only sequence that contains a 5,7,5-Haiku is [5, 7, 5].

Sample Input 2

4 5 7 5

Sample Output 2

34

Sample Input 3

37 4 2 3

Sample Output 3

863912418

Sample Input 4

40 5 7 5

Sample Output 4

562805100

Problem Statement

Iroha has a sequence of N strings s_1, s_2, ..., s_N.

She will choose some (possibly all) strings from the sequence, then concatenate those strings retaining the relative order, to produce a long string.

Among all strings of length K that she can produce in this way, find the lexicographically smallest one.

Constraints

• 1 ≦ N ≦ 2000
• 1 ≦ K ≦ 10^4
• For each i, 1 ≦ |s_i| ≦ K.
• |s_1| + |s_2| + ... + |s_N| ≦ 10^6
• For each i, s_i consists of lowercase letters.
• There exists at least one string of length K that Iroha can produce.

Sample Input 1

3 7
at
coder
codar

Sample Output 1

atcodar

at and codar should be chosen.

Sample Input 2

3 7
coder
codar
at

Sample Output 2

codarat

codar and at should be chosen.

Sample Input 3

4 13
kyuri
namida
zzzzzzz
aaaaaa

Sample Output 3

namidazzzzzzz

namida and zzzzzzz should be chosen.