Problem Statement

Snuke is making sugar water in a beaker. Initially, the beaker is empty. Snuke can perform the following four types of operations any number of times. He may choose not to perform some types of operations.

• Operation 1: Pour 100A grams of water into the beaker.
• Operation 2: Pour 100B grams of water into the beaker.
• Operation 3: Put C grams of sugar into the beaker.
• Operation 4: Put D grams of sugar into the beaker.

In our experimental environment, E grams of sugar can dissolve into 100 grams of water.

Snuke will make sugar water with the highest possible density.

The beaker can contain at most F grams of substances (water and sugar combined), and there must not be any undissolved sugar in the beaker. Find the mass of the sugar water Snuke will make, and the mass of sugar dissolved in it. If there is more than one candidate, any of them will be accepted.

We remind you that the sugar water that contains a grams of water and b grams of sugar is \frac{100b}{a + b} percent. Also, in this problem, pure water that does not contain any sugar is regarded as 0 percent density sugar water.

Constraints

• 1 \leq A < B \leq 30
• 1 \leq C < D \leq 30
• 1 \leq E \leq 100
• 100A \leq F \leq 3 000
• A, B, C, D, E and F are all integers.

Sample Input 1

1 2 10 20 15 200

Sample Output 1

110 10

In this environment, 15 grams of sugar can dissolve into 100 grams of water, and the beaker can contain at most 200 grams of substances.

We can make 110 grams of sugar water by performing Operation 1 once and Operation 3 once. It is not possible to make sugar water with higher density. For example, the following sequences of operations are infeasible:

• If we perform Operation 1 once and Operation 4 once, there will be undissolved sugar in the beaker.
• If we perform Operation 2 once and Operation 3 three times, the mass of substances in the beaker will exceed 200 grams.

Sample Input 2

1 2 1 2 100 1000

Sample Output 2

200 100

There are other acceptable outputs, such as:

400 200

However, the output below is not acceptable:

300 150

This is because, in order to make 300 grams of sugar water containing 150 grams of sugar, we need to pour exactly 150 grams of water into the beaker, which is impossible.

Sample Input 3

17 19 22 26 55 2802

Sample Output 3

2634 934

Problem Statement

In Takahashi Kingdom, which once existed, there are N cities, and some pairs of cities are connected bidirectionally by roads. The following are known about the road network:

• People traveled between cities only through roads. It was possible to reach any city from any other city, via intermediate cities if necessary.
• Different roads may have had different lengths, but all the lengths were positive integers.

Snuke the archeologist found a table with N rows and N columns, A, in the ruin of Takahashi Kingdom. He thought that it represented the shortest distances between the cities along the roads in the kingdom.

Determine whether there exists a road network such that for each u and v, the integer A_{u, v} at the u-th row and v-th column of A is equal to the length of the shortest path from City u to City v. If such a network exist, find the shortest possible total length of the roads.

Constraints

• 1 \leq N \leq 300
• If i ≠ j, 1 \leq A_{i, j} = A_{j, i} \leq 10^9.
• A_{i, i} = 0

Sample Input 1

3
0 1 3
1 0 2
3 2 0

Sample Output 1

3

The network below satisfies the condition:

• City 1 and City 2 is connected by a road of length 1.
• City 2 and City 3 is connected by a road of length 2.
• City 3 and City 1 is not connected by a road.

Sample Input 2

3
0 1 3
1 0 1
3 1 0

Sample Output 2

-1

As there is a path of length 1 from City 1 to City 2 and City 2 to City 3, there is a path of length 2 from City 1 to City 3. However, according to the table, the shortest distance between City 1 and City 3 must be 3.

Thus, we conclude that there exists no network that satisfies the condition.

Sample Input 3

5
0 21 18 11 28
21 0 13 10 26
18 13 0 23 13
11 10 23 0 17
28 26 13 17 0

Sample Output 3

82

Sample Input 4

3
0 1000000000 1000000000
1000000000 0 1000000000
1000000000 1000000000 0

Sample Output 4

3000000000

Problem Statement

We have a tree with N vertices. Vertex 1 is the root of the tree, and the parent of Vertex i (2 \leq i \leq N) is Vertex P_i.

To each vertex in the tree, Snuke will allocate a color, either black or white, and a non-negative integer weight.

Snuke has a favorite integer sequence, X_1, X_2, ..., X_N, so he wants to allocate colors and weights so that the following condition is satisfied for all v.

• The total weight of the vertices with the same color as v among the vertices contained in the subtree whose root is v, is X_v.

Here, the subtree whose root is v is the tree consisting of Vertex v and all of its descendants.

Determine whether it is possible to allocate colors and weights in this way.

Constraints

• 1 \leq N \leq 1 000
• 1 \leq P_i \leq i - 1
• 0 \leq X_i \leq 5 000

Sample Input 1

3
1 1
4 3 2

Sample Output 1

POSSIBLE

For example, the following allocation satisfies the condition:

• Set the color of Vertex 1 to white and its weight to 2.
• Set the color of Vertex 2 to black and its weight to 3.
• Set the color of Vertex 3 to white and its weight to 2.

There are also other possible allocations.

Sample Input 2

3
1 2
1 2 3

Sample Output 2

IMPOSSIBLE

If the same color is allocated to Vertex 2 and Vertex 3, Vertex 2 cannot be allocated a non-negative weight.

If different colors are allocated to Vertex 2 and 3, no matter which color is allocated to Vertex 1, it cannot be allocated a non-negative weight.

Thus, there exists no allocation of colors and weights that satisfies the condition.

Sample Input 3

8
1 1 1 3 4 5 5
4 1 6 2 2 1 3 3

Sample Output 3

POSSIBLE

Sample Input 4

1

0

Sample Output 4

POSSIBLE

Problem Statement

There are 2N balls in the xy-plane. The coordinates of the i-th of them is (x_i, y_i). Here, x_i and y_i are integers between 1 and N (inclusive) for all i, and no two balls occupy the same coordinates.

In order to collect these balls, Snuke prepared 2N robots, N of type A and N of type B. Then, he placed the type-A robots at coordinates (1, 0), (2, 0), ..., (N, 0), and the type-B robots at coordinates (0, 1), (0, 2), ..., (0, N), one at each position.

When activated, each type of robot will operate as follows.

• When a type-A robot is activated at coordinates (a, 0), it will move to the position of the ball with the lowest y-coordinate among the balls on the line x = a, collect the ball and deactivate itself. If there is no such ball, it will just deactivate itself without doing anything.

• When a type-B robot is activated at coordinates (0, b), it will move to the position of the ball with the lowest x-coordinate among the balls on the line y = b, collect the ball and deactivate itself. If there is no such ball, it will just deactivate itself without doing anything.

Once deactivated, a robot cannot be activated again. Also, while a robot is operating, no new robot can be activated until the operating robot is deactivated.

When Snuke was about to activate a robot, he noticed that he may fail to collect all the balls, depending on the order of activating the robots.

Among the (2N)! possible orders of activating the robots, find the number of the ones such that all the balls can be collected, modulo 1 000 000 007.

Constraints

• 2 \leq N \leq 10^5
• 1 \leq x_i \leq N
• 1 \leq y_i \leq N
• If i ≠ j, either x_i ≠ x_j or y_i ≠ y_j.

Sample Input 1

2
1 1
1 2
2 1
2 2

Sample Output 1

8

We will refer to the robots placed at (1, 0) and (2, 0) as A1 and A2, respectively, and the robots placed at (0, 1) and (0, 2) as B1 and B2, respectively. There are eight orders of activation that satisfy the condition, as follows:

• A1, B1, A2, B2
• A1, B1, B2, A2
• A1, B2, B1, A2
• A2, B1, A1, B2
• B1, A1, B2, A2
• B1, A1, A2, B2
• B1, A2, A1, B2
• B2, A1, B1, A2

Thus, the output should be 8.

Sample Input 2

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

Sample Output 2

7392

Sample Input 3

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

Sample Output 3

4480

Sample Input 4

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

Sample Output 4

82060779

Sample Input 5

3
1 1
1 2
1 3
2 1
2 2
2 3

Sample Output 5

0

When there is no order of activation that satisfies the condition, the output should be 0.