Problem Statement

There are N people standing in a line: person 1, person 2, \ldots, person N.

You are given the arrangement of the people as a sequence A=(A _ 1,A _ 2,\ldots,A _ N) of length N.

A _ i\ (1\leq i\leq N) represents the following information:

• if A _ i=-1, person i is at the front of the line;
• if A _ i\neq -1, person i is right behind person A _ i.

Print the people's numbers in the line from front to back.

Constraints

• 1\leq N\leq3\times10 ^ 5
• A _ i=-1 or 1\leq A _ i\leq N\ (1\leq i\leq N)
• There is exactly one way to arrange the N people consistent with the information given.
• All input values are integers.

Sample Input 1

6
4 1 -1 5 3 2

Sample Output 1

3 5 4 1 2 6

If person 3, person 5, person 4, person 1, person 2, and person 6 stand in line in this order from front to back, the arrangement matches the given information.

Indeed, it can be seen that:

• person 1 is standing right behind person 4,
• person 2 is standing right behind person 1,
• person 3 is at the front of the line,
• person 4 is standing right behind person 5,
• person 5 is standing right behind person 3, and
• person 6 is standing right behind person 2.

Thus, print 3, 5, 4, 1, 2, and 6 in this order, separated by spaces.

Sample Input 2

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

Sample Output 2

1 2 3 4 5 6 7 8 9 10

Sample Input 3

30
3 25 20 6 18 12 26 1 29 -1 21 17 23 9 8 30 10 15 22 27 4 13 5 11 16 24 28 2 19 7

Sample Output 3

10 17 12 6 4 21 11 24 26 7 30 16 25 2 28 27 20 3 1 8 15 18 5 23 13 22 19 29 9 14

Problem Statement

You have N keys numbered 1, 2, \dots, N.
Some of these are real keys, while the others are dummies.

There is a door, Door X, into which you can insert any number of keys. Door X will open if and only if at least K real keys are inserted.

You have conducted M tests on these keys. The i-th test went as follows:

• You inserted C_i keys A_{i,1}, A_{i,2}, \dots, A_{i,C_i} into Door X.
• The test result is represented by a single English letter R_i.
  • R_i = o means that Door X opened in the i-th test.
  • R_i = x means that Door X did not open in the i-th test.

There are 2^N possible combinations of which keys are real and which are dummies. Among these, find the number of combinations that do not contradict any of the test results.
It is possible that the given test results are incorrect and no combination satisfies the conditions. In such a case, report 0.

Constraints

• N, M, K, C_i, and A_{i,j} are integers.
• 1 \le K \le N \le 15
• 1 \le M \le 100
• 1 \le C_i \le N
• 1 \le A_{i,j} \le N
• A_{i,j} \neq A_{i,k} if j \neq k.
• R_i is o or x.

Sample Input 1

3 2 2
3 1 2 3 o
2 2 3 x

Sample Output 1

2

In this input, there are three keys and two tests were conducted.
Two correct keys are required to open Door X.

• In the first test, keys 1, 2, 3 were used, and Door X opened.
• In the second test, keys 2, 3 were used, and Door X did not open.

There are two combinations of which keys are real and which are dummies that do not contradict any of the test results:

• Key 1 is real, key 2 is a dummy, and key 3 is real.
• Key 1 is real, key 2 is real, and key 3 is a dummy.

Sample Input 2

4 5 3
3 1 2 3 o
3 2 3 4 o
3 3 4 1 o
3 4 1 2 o
4 1 2 3 4 x

Sample Output 2

0

As mentioned in the problem statement, the answer may be 0.

Sample Input 3

11 4 9
10 1 2 3 4 5 6 7 8 9 10 o
11 1 2 3 4 5 6 7 8 9 10 11 o
10 11 10 9 8 7 6 5 4 3 2 x
10 11 9 1 4 3 7 5 6 2 10 x

Sample Output 3

8

Problem Statement

There is a grid with H rows and W columns. Let (i, j) denote the cell at the i-th row from the top and j-th column from the left.
Initially, there is one wall in each cell.
After processing Q queries explained below in the order they are given, find the number of remaining walls.

In the q-th query, you are given two integers R_q and C_q.
You place a bomb at (R_q, C_q) to destroy walls. As a result, the following process occurs.

• If there is a wall at (R_q, C_q), destroy that wall and end the process.
• If there is no wall at (R_q, C_q), destroy the first walls that appear when looking up, down, left, and right from (R_q, C_q). More precisely, the following four processes occur simultaneously:
  • If there exists an i \lt R_q such that a wall exists at (i, C_q) and no wall exists at (k, C_q) for all i \lt k \lt R_q, destroy the wall at (i, C_q).
  • If there exists an i \gt R_q such that a wall exists at (i, C_q) and no wall exists at (k, C_q) for all R_q \lt k \lt i, destroy the wall at (i, C_q).
  • If there exists a j \lt C_q such that a wall exists at (R_q, j) and no wall exists at (R_q, k) for all j \lt k \lt C_q, destroy the wall at (R_q, j).
  • If there exists a j \gt C_q such that a wall exists at (R_q, j) and no wall exists at (R_q, k) for all C_q \lt k \lt j, destroy the wall at (R_q, j).

Constraints

• 1 \leq H, W
• H \times W \leq 4 \times 10^5
• 1 \leq Q \leq 2 \times 10^5
• 1 \leq R_q \leq H
• 1 \leq C_q \leq W
• All input values are integers.

Sample Input 1

2 4 3
1 2
1 2
1 3

Sample Output 1

2

The process of handling the queries can be explained as follows:

• In the 1st query, (R_1, C_1) = (1, 2). There is a wall at (1, 2), so the wall at (1, 2) is destroyed.
• In the 2nd query, (R_2, C_2) = (1, 2). There is no wall at (1, 2), so the walls at (2,2),(1,1),(1,3), which are the first walls that appear when looking up, down, left, and right from (1, 2), are destroyed.
• In the 3rd query, (R_3, C_3) = (1, 3). There is no wall at (1, 3), so the walls at (2,3),(1,4), which are the first walls that appear when looking up, down, left, and right from (1, 3), are destroyed.

After processing all queries, there are two remaining walls, at (2, 1) and (2, 4).

Sample Input 2

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

Sample Output 2

10

Sample Input 3

4 3 10
2 2
4 1
1 1
4 2
2 1
3 1
1 3
1 2
4 3
4 2

Sample Output 3

2

Problem Statement

You are given sequences of length N: A = (A_1, A_2, \dots, A_N) and B = (B_1, B_2, \dots, B_N).
Let S be a subset of \lbrace1, 2, \dots, N\rbrace of size K. Here, find the minimum possible value of the following expression:

You are given T test cases; solve each of them.

Constraints

• 1 \leq T \leq 2 \times 10^5
• 1 \leq K \leq N \leq 2 \times 10^5
• 1 \leq A_i, B_i \leq 10^6
• The sum of N over all test cases is at most 2 \times 10^5.
• All input values are integers.

Sample Input 1

3
3 2
3 7 6
9 2 4
5 3
6 4 1 5 9
8 6 5 1 7
10 6
61 95 61 57 69 49 46 47 14 43
39 79 48 92 90 76 30 16 30 94

Sample Output 1

42
60
14579

In the first test case, for S = \{2, 3\}, the value of the expression is 7 \times (2 + 4) = 42, which is the minimum.

Problem Statement

We have N dice. For each i = 1, 2, \ldots, N, when the i-th die is thrown, it shows a random integer between 1 and A_i, inclusive, with equal probability.

Find the probability, modulo 998244353, that the following condition is satisfied when the N dice are thrown simultaneously.

There is a way to choose some (possibly all) of the N dice so that the sum of their results is 10.

Constraints

• 1 \leq N \leq 100
• 1 \leq A_i \leq 10^6
• All input values are integers.

Sample Input 1

4
1 7 2 9

Sample Output 1

942786334

For instance, if the first, second, third, and fourth dice show 1, 3, 2, and 7, respectively, these results satisfy the condition. In fact, if the second and fourth dice are chosen, the sum of their results is 3 + 7 = 10. Alternatively, if the first, third, and fourth dice are chosen, the sum of their results is 1 + 2 + 7 = 10.

On the other hand, if the first, second, third, and fourth dice show 1, 6, 1, and 5, respectively, there is no way to choose some of them so that the sum of their results is 10, so the condition is not satisfied.

In this sample input, the probability of the results of the N dice satisfying the condition is \frac{11}{18}. Thus, print this value modulo 998244353, that is, 942786334.

Sample Input 2

7
1 10 100 1000 10000 100000 1000000

Sample Output 2

996117877