Problem Statement

This contest is CODEFESTIVAL, which can be shortened to the string CF by deleting some characters.

Mr. Takahashi, full of curiosity, wondered if he could obtain CF from other strings in the same way.

You are given a string s consisting of uppercase English letters. Determine whether the string CF can be obtained from the string s by deleting some characters.

Constraints

• 2 ≤ |s| ≤ 100
• All characters in s are uppercase English letters (A-Z).

Sample Input 1

CODEFESTIVAL

Sample Output 1

Yes

CF is obtained by deleting characters other than the first character C and the fifth character F.

Sample Input 2

FESTIVALCODE

Sample Output 2

No

FC can be obtained but CF cannot be obtained because you cannot change the order of the characters.

Sample Input 3

CF

Sample Output 3

Yes

It is also possible not to delete any characters.

Sample Input 4

FCF

Sample Output 4

Yes

CF is obtained by deleting the first character.

Problem Statement

There are K pieces of cakes. Mr. Takahashi would like to eat one cake per day, taking K days to eat them all.

There are T types of cake, and the number of the cakes of type i (1 ≤ i ≤ T) is a_i.

Eating the same type of cake two days in a row would be no fun, so Mr. Takahashi would like to decide the order for eating cakes that minimizes the number of days on which he has to eat the same type of cake as the day before.

Compute the minimum number of days on which the same type of cake as the previous day will be eaten.

Constraints

• 1 ≤ K ≤ 10000
• 1 ≤ T ≤ 100
• 1 ≤ a_i ≤ 100
• a_1 + a_2 + ... + a_T = K

Sample Input 1

7 3
3 2 2

Sample Output 1

0

For example, if Mr. Takahashi eats cakes in the order of 2, 1, 2, 3, 1, 3, 1, he can avoid eating the same type of cake as the previous day.

Sample Input 2

6 3
1 4 1

Sample Output 2

1

There are 6 cakes. For example, if Mr. Takahashi eats cakes in the order of 2, 3, 2, 2, 1, 2, he has to eat the same type of cake (i.e., type 2) as the previous day only on the fourth day. Since this is the minimum number, the answer is 1.

Sample Input 3

100 1
100

Sample Output 3

99

Since Mr. Takahashi has only one type of cake, he has no choice but to eat the same type of cake as the previous day from the second day and after.

Problem Statement

Mountaineers Mr. Takahashi and Mr. Aoki recently trekked across a certain famous mountain range. The mountain range consists of N mountains, extending from west to east in a straight line as Mt. 1, Mt. 2, ..., Mt. N. Mr. Takahashi traversed the range from the west and Mr. Aoki from the east.

The height of Mt. i is h_i, but they have forgotten the value of each h_i. Instead, for each i (1 ≤ i ≤ N), they recorded the maximum height of the mountains climbed up to the time they reached the peak of Mt. i (including Mt. i). Mr. Takahashi's record is T_i and Mr. Aoki's record is A_i.

We know that the height of each mountain h_i is a positive integer. Compute the number of the possible sequences of the mountains' heights, modulo 10^9 + 7.

Note that the records may be incorrect and thus there may be no possible sequence of the mountains' heights. In such a case, output 0.

Constraints

• 1 ≤ N ≤ 10^5
• 1 ≤ T_i ≤ 10^9
• 1 ≤ A_i ≤ 10^9
• T_i ≤ T_{i+1} (1 ≤ i ≤ N - 1)
• A_i ≥ A_{i+1} (1 ≤ i ≤ N - 1)

Sample Input 1

5
1 3 3 3 3
3 3 2 2 2

Sample Output 1

4

The possible sequences of the mountains' heights are:

• 1, 3, 2, 2, 2
• 1, 3, 2, 1, 2
• 1, 3, 1, 2, 2
• 1, 3, 1, 1, 2

for a total of four sequences.

Sample Input 2

5
1 1 1 2 2
3 2 1 1 1

Sample Output 2

0

The records are contradictory, since Mr. Takahashi recorded 2 as the highest peak after climbing all the mountains but Mr. Aoki recorded 3.

Sample Input 3

10
1 3776 3776 8848 8848 8848 8848 8848 8848 8848
8848 8848 8848 8848 8848 8848 8848 8848 3776 5

Sample Output 3

884111967

Don't forget to compute the number modulo 10^9 + 7.

Sample Input 4

1
17
17

Sample Output 4

1

Some mountain ranges consist of only one mountain.

Problem Statement

Mr. Takahashi has in his room an art object with H rows and W columns, made up of H \times W blocks. Each block has a color represented by a lowercase English letter (a-z). The color of the block at the i-th row and j-th column is c_{i,j}.

Mr. Takahashi would like to dismantle the object, finding it a bit kitschy for his tastes. The dismantling is processed by repeating the following operation:

• Choose one of the W columns and push down that column one row. The block at the bottom of that column disappears.

Each time the operation is performed, a cost is incurred. Since blocks of the same color have a property to draw each other together, the cost of the operation is the number of the pairs of blocks (p, q) such that:

• The block p is in the selected column.
• The two blocks p and q are horizontally adjacent (before pushing down the column).
• The two blocks p and q have the same color.

Mr. Takahashi dismantles the object by repeating the operation H \times W times to get rid of all the blocks. Compute the minimum total cost to dismantle the object.

Constraints

• 1 ≤ H ≤ 300
• 2 ≤ W ≤ 300
• All characters c_{i,j} are lowercase English letters (a-z).

Partial Score

• In test cases worth 300 points, W = 3.

Sample Input 1

2 3
rrr
brg

Sample Output 1

2

For example, the total cost of 2 can be achieved by performing the operation as follows and this is the minimum value.

Sample Input 2

6 3
xya
xya
ayz
ayz
xaz
xaz

Sample Output 2

0

The total cost of 0 can be achieved by first pushing down all blocks of the middle column, then all of the left column, and all of the right column.

Sample Input 3

4 2
ay
xa
xy
ay

Sample Output 3

0

The total cost of 0 can be achieved by the following operations:

• pushing down the right column one row;
• pushing down the left column one row;
• pushing down all of the right column;
• and pushing down all of the left column.

Sample Input 4

5 5
aaaaa
abbba
ababa
abbba
aaaaa

Sample Output 4

24

Sample Input 5

7 10
xxxxxxxxxx
ccccxxffff
cxxcxxfxxx
cxxxxxffff
cxxcxxfxxx
ccccxxfxxx
xxxxxxxxxx

Sample Output 5

130

Problem Statement

One day Mr. Takahashi picked up a dictionary containing all of the N! permutations of integers 1 through N. The dictionary has N! pages, and page i (1 ≤ i ≤ N!) contains the i-th permutation in the lexicographical order.

Mr. Takahashi wanted to look up a certain permutation of length N in this dictionary, but he forgot some part of it.

His memory of the permutation is described by a sequence P_1, P_2, ..., P_N. If P_i = 0, it means that he forgot the i-th element of the permutation; otherwise, it means that he remembered the i-th element of the permutation and it is P_i.

He decided to look up all the possible permutations in the dictionary. Compute the sum of the page numbers of the pages he has to check, modulo 10^9 + 7.

Constraints

• 1 ≤ N ≤ 500000
• 0 ≤ P_i ≤ N
• P_i ≠ P_j if i ≠ j (1 ≤ i, j ≤ N), P_i ≠ 0 and P_j ≠ 0.

Partial Score

• In test cases worth 500 points, 1 ≤ N ≤ 3000.

Sample Input 1

4
0 2 3 0

Sample Output 1

23

The possible permutations are [1, 2, 3, 4] and [4, 2, 3, 1]. Since the former is on page 1 and the latter is on page 22, the answer is 23.

Sample Input 2

3
0 0 0

Sample Output 2

21

Since all permutations of length 3 are possible, the answer is 1 + 2 + 3 + 4 + 5 + 6 = 21.

Sample Input 3

5
1 2 3 5 4

Sample Output 3

2

Mr. Takahashi remembered all the elements of the permutation.

Sample Input 4

1
0

Sample Output 4

1

The dictionary consists of one page.

Sample Input 5

10
0 3 0 0 1 0 4 0 0 0

Sample Output 5

953330050