Problem Statement

You are given N strings S_1,S_2,\dots,S_N, each of length M, consisting of lowercase English letter. Here, S_i are pairwise distinct.

Determine if one can rearrange these strings to obtain a new sequence of strings T_1,T_2,\dots,T_N such that:

• for all integers i such that 1 \le i \le N-1, one can alter exactly one character of T_i to another lowercase English letter to make it equal to T_{i+1}.

Constraints

• 2 \le N \le 8
• 1 \le M \le 5
• S_i is a string of length M consisting of lowercase English letters. (1 \le i \le N)
• S_i are pairwise distinct.

Sample Input 1

4 4
bbed
abcd
abed
fbed

Sample Output 1

Yes

One can rearrange them in this order: abcd, abed, bbed, fbed. This sequence satisfies the condition.

Sample Input 2

2 5
abcde
abced

Sample Output 2

No

No matter how the strings are rearranged, the condition is never satisfied.

Sample Input 3

8 4
fast
face
cast
race
fact
rice
nice
case

Sample Output 3

Yes

Problem Statement

Given an integer N, solve the following problem.

Let f(x)= (The number of positive integers at most x with the same number of digits as x).
Find f(1)+f(2)+\dots+f(N) modulo 998244353.

Constraints

• N is an integer.
• 1 \le N < 10^{18}

Sample Input 1

16

Sample Output 1

73

• For a positive integer x between 1 and 9, the positive integers at most x with the same number of digits as x are 1,2,\dots,x.
  • Thus, we have f(1)=1,f(2)=2,...,f(9)=9.
• For a positive integer x between 10 and 16, the positive integers at most x with the same number of digits as x are 10,11,\dots,x.
  • Thus, we have f(10)=1,f(11)=2,...,f(16)=7.

The final answer is 73.

Sample Input 2

238

Sample Output 2

13870

Sample Input 3

999999999999999999

Sample Output 3

762062362

Be sure to find the sum modulo 998244353.

Problem Statement

You are given a sequence of positive integers A=(a_1,\ldots,a_N) of length N.
There are (2^N-1) ways to choose one or more terms of A. How many of them have an integer-valued average? Find the count modulo 998244353.

Constraints

• 1 \leq N \leq 100
• 1 \leq a_i \leq 10^9
• All values in input are integers.

Sample Input 1

3
2 6 2

Sample Output 1

6

For each way to choose terms of A, the average is obtained as follows:

• If just a_1 is chosen, the average is \frac{a_1}{1}=\frac{2}{1} = 2, which is an integer.

• If just a_2 is chosen, the average is \frac{a_2}{1}=\frac{6}{1} = 6, which is an integer.

• If just a_3 is chosen, the average is \frac{a_3}{1}=\frac{2}{1} = 2, which is an integer.

• If a_1 and a_2 are chosen, the average is \frac{a_1+a_2}{2}=\frac{2+6}{2} = 4, which is an integer.

• If a_1 and a_3 are chosen, the average is \frac{a_1+a_3}{2}=\frac{2+2}{2} = 2, which is an integer.

• If a_2 and a_3 are chosen, the average is \frac{a_2+a_3}{2}=\frac{6+2}{2} = 4, which is an integer.

• If a_1, a_2, and a_3 are chosen, the average is \frac{a_1+a_2+a_3}{3}=\frac{2+6+2}{3} = \frac{10}{3}, which is not an integer.

Therefore, 6 ways satisfy the condition.

Sample Input 2

5
5 5 5 5 5

Sample Output 2

31

Regardless of the choice of one or more terms of A, the average equals 5.

Problem Statement

We have N books numbered 1 to N.
Book i assumes that you have read C_i books, the j-th of which is book P_{i,j}: you must read all these C_i books before reading book i.
Here, you can read all the books in some order.

You are trying to read the minimum number of books required to read book 1.
Print the numbers of the books you must read excluding book 1 in the order they should be read. Under this condition, the set of books to read is uniquely determined.
If there are multiple reading orders that satisfy the condition, you may print any of them.

Constraints

• 2 \leq N \leq 2 \times 10^5
• 0 \leq C_i < N
• \sum_{i=1}^{N} C_i \leq 2 \times 10^5
• C_1 \geq 1
• 1 \leq P_{i,j} \leq N
• P_{i,j} \neq P_{i,k} for 1 \leq j < k \leq C_i.
• It is possible to read all the books.

Sample Input 1

6
3 2 3 4
2 3 5
0
1 5
0
0

Sample Output 1

5 3 4 2

To read book 1, you must read books 2,3,4; to read book 2, you must read books 3,5; to read book 4, you must read book 5. To read books 3,5,6, you do not have to read any other books.

For example, if you read books 5,3,4,2 in this order, you can read book 1. This is a correct answer, because you will never be able to read book 1 with three or fewer books read. As another example, reading books 3,5,4,2 in this order also allows you to read book 1 with 4 books read.

Sample Input 2

6
1 2
1 3
1 4
1 5
1 6
0

Sample Output 2

6 5 4 3 2

Sample Input 3

8
1 5
1 6
1 7
1 8
0
0
0
0

Sample Output 3

5

Problem Statement

There are N cities numbered city 1, city 2, \ldots, and city N.
There are also one-way teleporters that send you to different cities. Whether a teleporter can send you directly from city i (1\leq i\leq N) to another is represented by a length-M string S_i consisting of 0 and 1. Specifically, for 1\leq j\leq N,

• if 1\leq j-i\leq M and the (j-i)-th character of S_i is 1, then a teleporter can send you directly from city i to city j;
• otherwise, it cannot send you directly from city i to city j.

Solve the following problem for k=2,3,\ldots, N-1:

Can you travel from city 1 to city N without visiting city k by repeatedly using a teleporter? If you can, print the minimum number of times you need to use a teleporter; otherwise, print -1.

Constraints

• 3 \leq N \leq 10^5
• 1\leq M\leq 10
• M<N
• S_i is a string of length M consisting of 0 and 1.
• If i+j>N, then the j-th character of S_i is 0.
• N and M are integers.

Sample Input 1

5 2
11
01
11
10
00

Sample Output 1

2 3 2

A teleporter sends you

• from city 1 to cities 2 and 3;
• from city 2 to city 4;
• from city 3 to cities 4 and 5;
• from city 4 to city 5; and
• from city 5 to nowhere.

Therefore, there are three paths to travel from city 1 to city 5:

• path 1 : city 1 \to city 2 \to city 4 \to city 5;
• path 2 : city 1 \to city 3 \to city 4 \to city 5; and
• path 3 : city 1 \to city 3 \to city 5.

Among these paths,

• two paths, path 2 and path 3, do not visit city 2. Among them, path 3 requires the minimum number of teleporter uses (twice).
• Path 1 is the only path without city 3. It requires using a teleporter three times.
• Path 3 is the only path without city 4. It requires using a teleporter twice.

Thus, 2, 3, and 2, separated by spaces, should be printed.

Sample Input 2

6 3
101
001
101
000
100
000

Sample Output 2

-1 3 3 -1

The only path from city 1 to city 6 is city 1 \to city 2 \to city 5 \to city 6.
For k=2,5, there is no way to travel from city 1 to city 6 without visiting city k.
For k=3,4, the path above satisfies the condition; it requires using a teleporter three times.

Thus, -1, 3, 3, and -1, separated by spaces, should be printed.

Note that a teleporter is one-way; a teleporter can send you from city 3 to city 4, but not from city 4 to city 3,
so the following path, for example, is invalid: city 1 \to city 4 \to city 3 \to city 6.