Problem Statement

There is a 10^9-story building with N ladders.
Takahashi, who is on the 1-st (lowest) floor, wants to reach the highest floor possible by using ladders (possibly none).
The ladders are numbered from 1 to N, and ladder i connects the A_i-th and B_i-th floors. One can use ladder i in either direction to move from the A_i-th floor to the B_i-th floor or vice versa, but not between other floors.
Takahashi can freely move within the same floor, but cannot move between floors without using ladders.
What is the highest floor Takahashi can reach?

Constraints

• 1 \leq N \leq 2 \times 10^5
• 1 \leq A_i, B_i \leq 10^9
• A_i \neq B_i
• All values in the input are integers.

Sample Input 1

4
1 4
4 3
4 10
8 3

Sample Output 1

10

He can reach the 10-th floor by using ladder 1 to get to the 4-th floor and then ladder 3 to get to the 10-th floor.

Sample Input 2

6
1 3
1 5
1 12
3 5
3 12
5 12

Sample Output 2

12

Sample Input 3

3
500000000 600000000
600000000 700000000
700000000 800000000

Sample Output 3

1

He may be unable to move between floors.

Problem Statement

Takahashi, who governs the Kingdom of AtCoder, has decided to change the alphabetical order of English lowercase letters.

The new alphabetical order is represented by a string X, which is a permutation of a, b, \ldots, z. The i-th character of X (1 \leq i \leq 26) would be the i-th smallest English lowercase letter in the new order.

The kingdom has N citizens, whose names are S_1, S_2, \dots, S_N, where each S_i (1 \leq i \leq N) consists of lowercase English letters.
Sort these names lexicographically according to the alphabetical order decided by Takahashi.

Constraints

• X is a permutation of a, b, \ldots, z.
• 2 \leq N \leq 50000
• N is an integer.
• 1 \leq |S_i| \leq 10 \, (1 \leq i \leq N)
• S_i consists of lowercase English letters. (1 \leq i \leq N)
• S_i \neq S_j (1 \leq i \lt j \leq N)

Sample Input 1

bacdefghijklmnopqrstuvwxzy
4
abx
bzz
bzy
caa

Sample Output 1

bzz
bzy
abx
caa

In the new alphabetical order set by Takahashi, b is smaller than a and z is smaller than y. Thus, sorting the citizens' names lexicographically would result in bzz, bzy, abx, caa in ascending order.

Sample Input 2

zyxwvutsrqponmlkjihgfedcba
5
a
ab
abc
ac
b

Sample Output 2

b
a
ac
ab
abc

Problem Statement

We have an undirected graph with (N_1+N_2) vertices and M edges. For i=1,2,\ldots,M, the i-th edge connects vertex a_i and vertex b_i.
The following properties are guaranteed:

• Vertex u and vertex v are connected, for all integers u and v with 1 \leq u,v \leq N_1.
• Vertex u and vertex v are connected, for all integers u and v with N_1+1 \leq u,v \leq N_1+N_2.
• Vertex 1 and vertex (N_1+N_2) are disconnected.

Consider performing the following operation exactly once:

• choose an integer u with 1 \leq u \leq N_1 and an integer v with N_1+1 \leq v \leq N_1+N_2, and add an edge connecting vertex u and vertex v.

We can show that vertex 1 and vertex (N_1+N_2) are always connected in the resulting graph; so let d be the minimum length (number of edges) of a path between vertex 1 and vertex (N_1+N_2).

Find the maximum possible d resulting from adding an appropriate edge to add.

Constraints

• 1 \leq N_1,N_2 \leq 1.5 \times 10^5
• 0 \leq M \leq 3 \times 10^5
• 1 \leq a_i \leq b_i \leq N_1+N_2
• (a_i,b_i) \neq (a_j,b_j) if i \neq j.
• Vertex u and vertex v are connected for all integers u and v such that 1 \leq u,v \leq N_1.
• Vertex u and vertex v are connected for all integers u and v such that N_1+1 \leq u,v \leq N_1+N_2.
• Vertex 1 and vertex (N_1+N_2) are disconnected.
• All input values are integers.

Sample Input 1

3 4 6
1 2
2 3
4 5
4 6
1 3
6 7

Sample Output 1

5

If we set u=2 and v=5, the operation yields d=5, which is the maximum possible.

Sample Input 2

7 5 20
10 11
4 5
10 12
1 2
1 5
5 6
2 4
3 5
9 10
2 5
1 4
11 12
9 12
8 9
5 7
3 7
3 6
3 4
8 12
9 11

Sample Output 2

4

Problem Statement

There are N cities. There are also one-way direct flights that connect different cities.
The availability of direct flights is represented by N strings S_1,S_2,\ldots,S_N of length N each. If the j-th character of S_i is Y, there is a direct flight from city i to city j; if it is N, there is not.
Each city sells a souvenir; city i sells a souvenir of value A_i.

Consider the following problem:

Takahashi is currently at city S and wants to get to city T (that is different from city S) using some direct flights.
Every time he visits a city (including S and T), he buys a souvenir there.
If there are multiple routes from city S to city T, Takahashi decides the route as follows:

• He tries to minimize the number of direct flights in the route from city S to city T.
• Then he tries to maximize the total value of the souvenirs he buys.

Determine if he can travel from city S to city T using the direct flights. If he can, find the "number of direct flights" and "total value of souvenirs" in the route that satisfies the conditions above.

You are given Q pairs (U_i,V_i) of distinct cities.
For each 1\leq i\leq Q, print the answer to the problem above when S=U_i and T=V_i.

Constraints

• 2 \leq N \leq 300
• 1\leq A_i\leq 10^9
• S_i is a string of length N consisting of Y and N.
• The i-th character of S_i is N.
• 1\leq Q\leq N(N-1)
• 1\leq U_i,V_i\leq N
• U_i\neq V_i
• If i \neq j, then (U_i,V_i)\neq (U_j,V_J).
• N,A_i,Q,U_i, and V_i are all integers.

Sample Input 1

5
30 50 70 20 60
NYYNN
NNYNN
NNNYY
YNNNN
YNNNN
3
1 3
3 1
4 5

Sample Output 1

1 100
2 160
3 180

For (S,T)=(U_1,V_1)=(1,3), there is a direct flight from city 1 to city 3, so the minimum possible number of direct flights is 1, which is achieved when he uses that direct flight. In this case, the total value of the souvenirs is A_1+A_3=30+70=100.

For (S,T)=(U_2,V_2)=(3,1), the minimum possible number of direct flights is 2. The following two routes achieve the minimum: cities 3\to 4\to 1, and cities 3\to 5\to 1. Since the total value of souvenirs in the two routes are 70+20+30=120 and 70+60+30=160, respectively, so he chooses the latter route, and the total value of the souvenirs is 160.

For (S,T)=(U_3,V_3)=(4,5), the number of direct flights is minimum when he travels cities 4\to 1\to 3\to 5, where the total value of the souvenirs is 20+30+70+60=180.

Sample Input 2

2
100 100
NN
NN
1
1 2

Sample Output 2

Impossible

There may be no direct flight at all.

Problem Statement

There are N people standing in a line, with person i standing at the i-th position from the front.

Repeat the following operation until there is only one person left in the line:

• Remove the person at the front of the line with a probability of \frac{1}{2}, otherwise move them to the end of the line.

For each person i=1,2,\ldots,N, find the probability that person i is the last person remaining in the line, modulo 998244353. (All choices to remove or not are random and independent.)

Constraints

• 2\leq N\leq 3000
• All input values are integers.

Sample Input 1

2

Sample Output 1

332748118 665496236

Person 1 will be the last person remaining in the line with probability \frac{1}{3}.

Person 2 will be the last person remaining in the line with probability \frac{2}{3}.

Sample Input 2

5

Sample Output 2

235530465 792768557 258531487 238597268 471060930