Problem Statement

We have N boxes numbered 1 to N that are initially empty, and an unlimited number of blank cards.
Process Q queries in order. There are three kinds of queries as follows.

• 1 i j \colon Write the number i on a blank card and put it into box j.
• 2 i \colon Report all numbers written on the cards in box i, in ascending order.
• 3 i \colon Report all box numbers of the boxes that contain a card with the number i, in ascending order.

Here, note the following.

• In a query of the second kind, if box i contains multiple cards with the same number, that number should be printed the number of times equal to the number of those cards.
• In a query of the third kind, even if a box contains multiple cards with the number i, the box number of that box should be printed only once.

Constraints

• 1 \leq N, Q \leq 2 \times 10^5
• For a query of the first kind:
  • 1 \leq i \leq 2 \times 10^5
  • 1 \leq j \leq N
• For a query of the second kind:
  • 1 \leq i \leq N
  • Box i contains some cards when this query is given.
• For a query of the third kind:
  • 1 \leq i \leq 2 \times 10^5
  • Some boxes contain a card with the number i when this query is given.
• At most 2 \times 10^5 numbers are to be reported.
• All values in the input are integers.

Sample Input 1

5
8
1 1 1
1 2 4
1 1 4
2 4
1 1 4
2 4
3 1
3 2

Sample Output 1

1 2
1 1 2
1 4
4

Let us process the queries in order.

• Write 1 on a card and put it into box 1.
• Write 2 on a card and put it into box 4.
• Write 1 on a card and put it into box 4.
• Box 4 contains cards with the numbers 1 and 2.
  • Print 1 and 2 in this order.
• Write 1 on a card and put it into box 4.
• Box 4 contains cards with the numbers 1, 1, and 2.
  • Note that you should print 1 twice.
• Boxes 1 and 4 contain a card with the number 1.
  • Note that you should print 4 only once, even though box 4 contains two cards with the number 1.
• Boxes 4 contains a card with the number 2.

Sample Input 2

1
5
1 1 1
1 2 1
1 200000 1
2 1
3 200000

Sample Output 2

1 2 200000
1

Problem Statement

You are given patterns S and T consisting of # and ., each with H rows and W columns.
The pattern S is given as H strings, and the j-th character of S_i represents the element at the i-th row and j-th column. The same goes for T.

Determine whether S can be made equal to T by rearranging the columns of S.

Here, rearranging the columns of a pattern X is done as follows.

• Choose a permutation P=(P_1,P_2,\dots,P_W) of (1,2,\dots,W).
• Then, for every integer i such that 1 \le i \le H, simultaneously do the following.
  • For every integer j such that 1 \le j \le W, simultaneously replace the element at the i-th row and j-th column of X with the element at the i-th row and P_j-th column.

Constraints

• H and W are integers.
• 1 \le H,W
• 1 \le H \times W \le 4 \times 10^5
• S_i and T_i are strings of length W consisting of # and ..

Sample Input 1

3 4
##.#
##..
#...
.###
..##
...#

Sample Output 1

Yes

If you, for instance, arrange the 3-rd, 4-th, 2-nd, and 1-st columns of S in this order from left to right, S will be equal to T.

Sample Input 2

3 3
#.#
.#.
#.#
##.
##.
.#.

Sample Output 2

No

In this input, S cannot be made equal to T.

Sample Input 3

2 1
#
.
#
.

Sample Output 3

Yes

It is possible that S=T.

Sample Input 4

8 7
#..#..#
.##.##.
#..#..#
.##.##.
#..#..#
.##.##.
#..#..#
.##.##.
....###
####...
....###
####...
....###
####...
....###
####...

Sample Output 4

Yes

Problem Statement

Takahashi and Aoki will play a game of taking stones using a sequence (A_1, \ldots, A_K).

There is a pile that initially contains N stones. The two players will alternately perform the following operation, with Takahashi going first.

• Choose an A_i that is at most the current number of stones in the pile. Remove A_i stones from the pile.

The game ends when the pile has no stones.

If both players attempt to maximize the total number of stones they remove before the end of the game, how many stones will Takahashi remove?

Constraints

• 1 \leq N \leq 10^4
• 1 \leq K \leq 100
• 1 = A_1 < A_2 < \ldots < A_K \leq N
• All values in the input are integers.

Sample Input 1

10 2
1 4

Sample Output 1

5

Below is one possible progression of the game.

• Takahashi removes 4 stones from the pile.
• Aoki removes 4 stones from the pile.
• Takahashi removes 1 stone from the pile.
• Aoki removes 1 stone from the pile.

In this case, Takahashi removes 5 stones. There is no way for him to remove 6 or more stones, so this is the maximum.

Below is another possible progression of the game where Takahashi removes 5 stones.

• Takahashi removes 1 stone from the pile.
• Aoki removes 4 stones from the pile.
• Takahashi removes 4 stones from the pile.
• Aoki removes 1 stone from the pile.

Sample Input 2

11 4
1 2 3 6

Sample Output 2

8

Below is one possible progression of the game.

• Takahashi removes 6 stones.
• Aoki removes 3 stones.
• Takahashi removes 2 stones.

In this case, Takahashi removes 8 stones. There is no way for him to remove 9 or more stones, so this is the maximum.

Sample Input 3

10000 10
1 2 4 8 16 32 64 128 256 512

Sample Output 3

5136

Problem Statement

We have a sequence A=(A_1,A_2,\dots,A_N) of length N. Initially, all the terms are 0.
Using an integer K given in the input, we define a function f(A) as follows:

• Let B be the sequence obtained by sorting A in descending order (so that it becomes monotonically non-increasing).
• Then, let f(A)=B_1 + B_2 + \dots + B_K.

We consider applying Q updates on this sequence.
Apply the following operation on the sequence A for i=1,2,\dots,Q in this order, and print the value f(A) at that point after each update.

• Change A_{X_i} to Y_i.

Constraints

• All input values are integers.
• 1 \le K \le N \le 5 \times 10^5
• 1 \le Q \le 5 \times 10^5
• 1 \le X_i \le N
• 0 \le Y_i \le 10^9

Sample Input 1

4 2 10
1 5
2 1
3 3
4 2
2 10
1 0
4 0
3 1
2 0
3 0

Sample Output 1

5
6
8
8
15
13
13
11
1
0

In this input, N=4 and K=2. Q=10 updates are applied.

• The 1-st update makes A=(5, 0,0,0). Now, f(A)=5.
• The 2-nd update makes A=(5, 1,0,0). Now, f(A)=6.
• The 3-rd update makes A=(5, 1,3,0). Now, f(A)=8.
• The 4-th update makes A=(5, 1,3,2). Now, f(A)=8.
• The 5-th update makes A=(5,10,3,2). Now, f(A)=15.
• The 6-th update makes A=(0,10,3,2). Now, f(A)=13.
• The 7-th update makes A=(0,10,3,0). Now, f(A)=13.
• The 8-th update makes A=(0,10,1,0). Now, f(A)=11.
• The 9-th update makes A=(0, 0,1,0). Now, f(A)=1.
• The 10-th update makes A=(0, 0,0,0). Now, f(A)=0.

Problem Statement

You are given a sequence A = (A_1, A_2, \dots, A_N) of length N.

Answer Q queries. The i-th query (1 \leq i \leq Q) is as follows:

• You are given integers R_i and X_i. Consider a subsequence (not necessarily contiguous) of (A_1, A_2, \dots, A_{R_i}) that is strictly increasing and consists only of elements at most X_i. Find the maximum possible length of such a subsequence. It is guaranteed that X_i \geq \min\lbrace A_1, A_2,\dots,A_{R_i} \rbrace.

Constraints

• 1 \leq N,Q \leq 2 \times 10^5
• 1 \leq A_i \leq 10^9
• 1 \leq R_i \leq N
• \min\lbrace A_1, A_2,\dots,A_{R_i} \rbrace\leq X_i\leq 10^9
• All input values are integers.

Sample Input 1

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

Sample Output 1

2
1
2

• 1st query: For the sequence (2,4), the longest strictly increasing subsequence with all elements at most 5 has length 2.
  Specifically, (2,4) qualifies.
• 2nd query: For the sequence (2,4,1,3,3), the longest strictly increasing subsequence with all elements at most 2 has length 1.
  Specifically, (2) and (1) qualify.
• 3rd query: For the sequence (2,4,1,3,3), the longest strictly increasing subsequence with all elements at most 3 has length 2.
  Specifically, (2,3) and (1,3) qualify.

Sample Input 2

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

Sample Output 2

4
1
1
2
1
5
3
4