Problem Statement

There are M sets, called S_1, S_2, \dots, S_M, consisting of integers between 1 and N.
S_i consists of C_i integers a_{i, 1}, a_{i, 2}, \dots, a_{i, C_i}.

There are (2^M-1) ways to choose one or more sets from the M sets.
How many of them satisfy the following condition?

• For all integers x such that 1 \leq x \leq N, there is at least one chosen set containing x.

Constraints

• 1 \leq N \leq 10
• 1 \leq M \leq 10
• 1 \leq C_i \leq N
• 1 \leq a_{i,1} \lt a_{i,2} \lt \dots \lt a_{i,C_i} \leq N
• All values in the input are integers.

Sample Input 1

3 3
2
1 2
2
1 3
1
2

Sample Output 1

3

The sets given in the input are S_1 = \lbrace 1, 2 \rbrace, S_2 = \lbrace 1, 3 \rbrace, S_3 = \lbrace 2 \rbrace.
The following three ways satisfy the conditions in the Problem Statement:

• choosing S_1, S_2;
• choosing S_1, S_2, S_3;
• choosing S_2, S_3.

Sample Input 2

4 2
2
1 2
2
1 3

Sample Output 2

0

There may be no way to choose sets that satisfies the conditions in the Problem Statement.

Sample Input 3

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

Sample Output 3

18

Problem Statement

A region has N towns numbered 1 to N, and M roads numbered 1 to M.

The i-th road connects town A_i and town B_i bidirectionally with length C_i.

Find the maximum possible total length of the roads you traverse when starting from a town of your choice and getting to another town without passing through the same town more than once.

Constraints

• 2 \leq N \leq 10
• 1 \leq M \leq \frac{N(N-1)}{2}
• 1 \leq A_i < B_i \leq N
• The pairs (A_i,B_i) are distinct.
• 1\leq C_i \leq 10^8
• All input values are integers.

Sample Input 1

4 4
1 2 1
2 3 10
1 3 100
1 4 1000

Sample Output 1

1110

If you travel as 4\to 1\to 3\to 2, the total length of the roads you traverse is 1110.

Sample Input 2

10 1
5 9 1

Sample Output 2

1

There may be a town that is not connected to a road.

Sample Input 3

10 13
1 2 1
1 10 1
2 3 1
3 4 4
4 7 2
4 8 1
5 8 1
5 9 3
6 8 1
6 9 5
7 8 1
7 9 4
9 10 3

Sample Output 3

20

Problem Statement

There is a sequence A = (A_0, A_1, \dots, A_{N - 1}) with N = 2^{20} terms. Initially, every term is -1.

Process Q queries in order. The i-th query (1 \leq i \leq Q) is described by an integer t_i such that t_i = 1 or t_i = 2, and another integer x_i, as follows.

• If t_i = 1, do the following in order.
  1. Define an integer h as h = x_i.
  2. While A_{h \bmod N} \neq -1, keep adding 1 to h. We can prove that this process ends after finite iterations under the Constraints of this problem.
  3. Replace the value of A_{h \bmod N} with x_i.
• If t_i = 2, print the value of A_{x_i \bmod N} at that time.

Here, for integers a and b, a \bmod b denotes the remainder when a is divided by b.

Constraints

• 1 \leq Q \leq 2 \times 10^5
• t_i \in \{ 1, 2 \} \, (1 \leq i \leq Q)
• 0 \leq x_i \leq 10^{18} \, (1 \leq i \leq Q)
• There is at least one i (1 \leq i \leq Q) such that t_i = 2.
• All values in input are integers.

Sample Input 1

4
1 1048577
1 1
2 2097153
2 3

Sample Output 1

1048577
-1

We have x_1 \bmod N = 1, so the first query sets A_1 = 1048577.

In the second query, initially we have h = x_2, for which A_{h \bmod N} = A_{1} \neq -1, so we add 1 to h. Now we have A_{h \bmod N} = A_{2} = -1, so this query sets A_2 = 1.

In the third query, we print A_{x_3 \bmod N} = A_{1} = 1048577.

In the fourth query, we print A_{x_4 \bmod N} = A_{3} = -1.

Note that, in this problem, N = 2^{20} = 1048576 is a constant and not given in input.

Problem Statement

Takahashi and Aoki are playing a game using N cards. The front side of the i-th card has A_i written on it, and the back side has B_i written on it. Initially, the N cards are laid out on the table. With Takahashi going first, the two players take turns performing the following operation:

• Choose a pair of cards from the table such that either the numbers on their front sides are the same or the numbers on their back sides are the same, and remove these two cards from the table. If no such pair of cards exists, the player cannot perform the operation.

The player who is first to be unable to perform the operation loses, and the other player wins. Determine who wins if both players play optimally.

Constraints

• 1 \leq N \leq 18
• 1 \leq A_i, B_i \leq 10^9
• All input values are integers.

Sample Input 1

5
1 9
2 5
4 9
1 4
2 5

Sample Output 1

Aoki

If Takahashi first removes

• the first and third cards: Aoki can win by removing the second and fifth cards.

• the first and fourth cards: Aoki can win by removing the second and fifth cards.

• the second and fifth cards: Aoki can win by removing the first and third cards.

These are the only three pairs of cards Takahashi can remove in his first move, and Aoki can win in all cases. Therefore, the answer is Aoki.

Sample Input 2

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

Sample Output 2

Takahashi

Problem Statement

There are N towns numbered 1,\ldots,N and M roads numbered 1,\ldots,M.

Road i connects towns A_i and B_i. When you use a road, your score changes as follows:

• when you move from town A_i to town B_i using road i, your score increases by C_i; when you move from town B_i to town A_i using road i, your score decreases by C_i.

Your score may become negative.

Answer the following Q questions.

• If you start traveling from town X_i with initial score 0, find the maximum possible score when you are at town Y_i.
  Here, if you cannot get from town X_i to town Y_i, print nan instead; if you can have as large a score as you want when you are at town Y_i, print inf instead.

Constraints

• 2\leq N \leq 10^5
• 0\leq M \leq 10^5
• 1\leq Q \leq 10^5
• 1\leq A_i,B_i,X_i,Y_i \leq N
• 0\leq C_i \leq 10^9
• All values in the input are integers.

Sample Input 1

5 5 3
1 2 1
1 2 2
3 4 1
4 5 1
3 5 2
5 3
1 2
3 1

Sample Output 1

-2
inf
nan

For the first question, if you use road 5 to move from town 5 to town 3, you can have a score -2 when you are at town 3.
Since you cannot make the score larger, the answer is -2.

For the second question, you can have as large a score as you want when you are at town 2 if you travel as follows: repeatedly "use road 2 to move from town 1 to town 2 and then use road 1 to move from town 2 to town 1" as many times as you want, and finally use road 2 to move from town 1 to town 2.

For the third question, you cannot get from town 3 to town 1.

Sample Input 2

2 1 1
1 1 1
1 1

Sample Output 2

inf

The endpoints of a road may be the same, and so may the endpoints given in a question.

Sample Input 3

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

Sample Output 3

inf
nan
nan
inf
-9