Problem Statement

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

Find the maximum value of \displaystyle \sum_{i=1}^{M} i \times B_i for a contiguous subarray B=(B_1,B_2,\dots,B_M) of A of length M.

Notes

A contiguous subarray of a number sequence is a sequence that is obtained by removing 0 or more initial terms and 0 or more final terms from the original number sequence.

For example, (2, 3) and (1, 2, 3) are contiguous subarrays of (1, 2, 3, 4), but (1, 3) and (3,2,1) are not contiguous subarrays of (1, 2, 3, 4).

Constraints

• 1 \le M \le N \le 2 \times 10^5
• - 2 \times 10^5 \le A_i \le 2 \times 10^5
• All values in input are integers.

Sample Input 1

4 2
5 4 -1 8

Sample Output 1

15

When B=(A_3,A_4), we have \displaystyle \sum_{i=1}^{M} i \times B_i = 1 \times (-1) + 2 \times 8 = 15. Since it is impossible to achieve 16 or a larger value, the solution is 15.

Note that you are not allowed to choose, for instance, B=(A_1,A_4).

Sample Input 2

10 4
-3 1 -4 1 -5 9 -2 6 -5 3

Sample Output 2

31

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

We have a video game consisting of N stages. The i-th stage (1 \leq i \leq N) is composed of a movie lasting A_i minutes and gameplay lasting B_i minutes.

To clear the i-th stage for the first time, one must watch the movie and do the gameplay for that stage. For the second and subsequent times, one may skip the movie and do just the gameplay.

In the beginning, only the 1-st stage is unlocked, and clearing the i-th stage (1 \leq i \leq N - 1) unlocks the (i+1)-th stage.

Find the shortest time needed to clear a stage X times in total. Here, if the same stage is cleared multiple times, all of them count.

Constraints

• 1 \leq N \leq 2 \times 10^5
• 1 \leq A_i, B_i \leq 10^9 \, (1 \leq i \leq N)
• 1 \leq X \leq 10^9
• All values in input are integers.

Sample Input 1

3 4
3 4
2 3
4 2

Sample Output 1

18

Here is one way to clear a stage 4 times in 18 minutes:

• Clear Stage 1. It takes A_1 + B_1 = 7 minutes.
• Clear Stage 2. It takes A_2 + B_2 = 5 minutes.
• Clear Stage 2 again. It takes B_2= 3 minutes.
• Clear Stage 2 again. It takes B_2= 3 minutes.

It is impossible to clear a stage 4 times within 17 minutes.

Sample Input 2

10 1000000000
3 3
1 6
4 7
1 8
5 7
9 9
2 4
6 4
5 1
3 1

Sample Output 2

1000000076

Problem Statement

Let us call a positive integer n that satisfies the following condition an arithmetic number.

• Let d_i be the i-th digit of n from the top (when n is written in base 10 without unnecessary leading zeros.) Then, (d_2-d_1)=(d_3-d_2)=\dots=(d_k-d_{k-1}) holds, where k is the number of digits in n.
  • This condition can be rephrased into the sequence (d_1,d_2,\dots,d_k) being arithmetic.
  • If n is a 1-digit integer, it is assumed to be an arithmetic number.

For example, 234,369,86420,17,95,8,11,777 are arithmetic numbers, while 751,919,2022,246810,2356 are not.

Find the smallest arithmetic number not less than X.

Constraints

• X is an integer between 1 and 10^{17} (inclusive).

Sample Input 1

152

Sample Output 1

159

The smallest arithmetic number not less than 152 is 159.

Sample Input 2

88

Sample Output 2

88

X itself may be an arithmetic number.

Sample Input 3

8989898989

Sample Output 3

9876543210

Problem Statement

Takahashi has an integer x. Initially, x = 0.

There are N operations. The i-th operation (1 \leq i \leq N) is represented by two integers t_i and y_i as follows:

• If t_i = 1, replace x with y_i.
• If t_i = 2, replace x with x + y_i.

Takahashi may skip any number between 0 and K (inclusive) of the operations. When he performs the remaining operations once each without changing the order, find the maximum possible final value of x.

Constraints

• 1 \leq N \leq 2 \times 10^5
• 0 \leq K \leq N
• t_i \in \{1,2\} \, (1 \leq i \leq N)
• |y_i| \leq 10^9 \, (1 \leq i \leq N)
• All values in input are integers.

Sample Input 1

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

Sample Output 1

3

If he skips the 5-th operation, x changes as 0 \rightarrow 4 \rightarrow 1 \rightarrow 2 \rightarrow 3, so x results in 3. This is the maximum.

Sample Input 2

1 0
2 -1000000000

Sample Output 2

-1000000000

Sample Input 3

10 3
2 3
2 -1
1 4
2 -1
2 5
2 -9
2 2
1 -6
2 5
2 -3

Sample Output 3

15