Description

Chocolate in its many forms is enjoyed by millions of people around the world every day. It is a truly universal candy available in virtually every country around the world.

You find that the only thing better than eating chocolate is to share it with friends. Unfortunately your friends are very picky and have different appetites: some would like more and others less of the chocolate that you offer them. You have found it increasingly difficult to determine whether their demands can be met. It is time to writte a program that solves the problem once and for all!

Your chocolate comes as a rectangular bar. The bar consists of same-sized rectangular pieces. To share the chocolate you may break one bar into two pieces along a division between rows or columns of the bar. You may then repeatedly break the resulting pieces in the same manner. Each of your friends insists on a getting a single rectangular portion of the chocolate that has a specified number of pieces. You are a little bit insistent as well: you will break up your bar only if all of it can be distributed to your friends, with none left over.

For exampla, Figure 9 shows one way that a chocolate bar consisting of 3 × 4 pieces can be split into 4 parts that contain 6, 3, 2, and 1 pieces respectively, by breanking it 3 times (This corresponds to the first sample input.)

Input

The input consists of multiple test cases each describing a chocolate bar to share. Each description starts with a line containing a single integer n (1 ≤ n ≤ 15), the number of parts in which the bar is supposed to be split. This is followed by a line containing two integers x and y (1 ≤ x, y ≤ 100), the dimensions of the chocolate bar. The next line contains n positive integers, giving the number of pieces that are supposed to be in each of the n parts. The input is terminated by a line containing the integer zero.

Output

For each test case, first display its case number. Then display whether it is possible to break the chocolate in the desired way: display ``Yes" if it is possible, and ``No" otherwise. Follow the format of the sample output.

Sample Input  Download

Sample Output  Download

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Description

Somewhere near the south pole, a number of penguins are standing on a number of ice floes. Being social animals, the penguins would like to get together, all on the same floe. The penguins do not want to get wet, so they have use their limited jump distance to get together by jumping from piece to piece. However, temperatures have been high lately, and the floes are showing cracks, and they get damaged further by the force needed to jump to another floe. Fortunately the penguins are real experts on cracking ice floes, and know exactly how many times a penguin can jump off each floe before it disintegrates and disappears. Landing on an ice floe does not damage it. You have to help the penguins find all floes where they can meet.

A sample layout of ice floes with 3 penguins on them.

Input

On the first line one positive number: the number of testcases, at most 100. After that per testcase:

• One line with the integer N (1 ≤ N ≤ 100) and a floating-point number D (0 ≤ D ≤ 100000), denoting the number of ice pieces and the maximum distance a penguin can jump.
• N lines, each line containing x_i, y_i, n_i and m_i, denoting for each ice piece its X and Y coordinate, the number of penguins on it and the maximum number of times a penguin can jump off this piece before it disappears (-10000 ≤ x_i, y_i ≤ 10000, 0 ≤ n_i ≤ 10, 1 ≤ m_i ≤ 200).

Output

Per testcase:

• One line containing a space-separated list of 0-based indices of the pieces on which all penguins can meet. If no such piece exists, output a line with the single number -1.

Sample Input  Download

Sample Output  Download

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Description

Given a connected, undirected graph G = (V, E), where V is the vertex set consisting a collection of nodes, and E is the set of edges, each of which connects two nodes from V. A vertex subset S is a separator if the subgraph induced by the vertices in V, but not in S, has two connected components. We shall use the notation [S, W, B] to represent the partition, where the removal of the separator S will give two connected components W and B.

In this problem, we consider the separators in grids. Each node in a grid is connected to its eight neighbors (if they exist). In Figure-1, we illustrate a partition of a 6*6 grid with a 9-point separator (gray nodes in the figure). The nodes on the left of the separator are in set W (white nodes), and the nodes on the right of the separator are in set B (black nodes).

To simplify the problem, you can assume that all the separators referred in this problem satisfy the following restrictions:

1) It’s a minimal separator. A separator is minimal if no subset of it forms a separator.
2) It begins from a node on the top line of the grid, except the corner (i.e. 30 and 35 in the figures), and ends with a node on the bottom line of the grid, also except the corner (i.e. 0 and 5 in the figures).
3) On its way from top to bottom, it can go left, right or down, but never go up.

Now we describe a method to improve a given partition on a grid, through which we can reduce the number of nodes in the separator. This method contains two steps:

1) Select several nodes from B and add them into S. Any of the selected nodes must have a left neighbor which is in S.
2) Remove several nodes from S (excluding the nodes added in the former step), and add them into W.

After the improvement, we should ensure S is still a separator, and make the number of nodes in S as small as possible. As for Figure-1, we should add 14 and 20 into S, and remove 7, 13, 19 and 25 from S. After that, we obtain a new partition with a 7-point separator shown in Figure-2.

Your task is, given a partition on a grid, to determine the least number of nodes in the separator after the improvement.

Input

There are several test cases. Each case begins with a line containing two integers, N and M (3 ≤ M, N ≤ 200). In each of the following N lines, there are M characters, describing the initial partition of the M*N grid. Every character is ‘S’, ‘W’ or ‘B’. It is confirmed that each of these three characters appears at least once in each line, and ‘W’s are always on the left of ‘S’s.

A test case of N = 0 and M = 0 indicates the end of input, and should not be processed.

Output

For each test case, you should output one line containing one integer, which is the least number of nodes in the separator after the improvement.

Sample Input  Download

Sample Output  Download

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