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A Strange Tree (S-tree) over the variable set X_{n} = {x_{1}, x_{2}, ..., x_{n}} is a binary tree representing a Boolean function f : {0, 1}^{n} → {0, 1}. Each path of the S-tree begins at the root node and consists of n+1 nodes. Each of the S-tree's nodes has a depth, which is the amount of nodes between itself and the root (So the root has depth 0). The nodes with depth less than n are called non-terminal nodes. All non-terminal nodes have two children: the right child and the left child. Each non-terminal node is marked with some variable x, from the variable set X_{n}. All non-terminal nodes with the same depth are marked with the same variable, and non-terminal nodes with different depth are marked with different variables. So, there is a unique variable x_{i1} corresponding to the root, a unique variable x_{i2} corresponding to the nodes with depth 1, and so on. The sequence of the variables x_{i1}, x_{i2}, ..., x_{in} is called the variable ordering. The nodes having depth n are called terminal nodes. They have no children and are marked with either 0 or 1. Note that the variable ordering and the distribution of 0's and 1's on terminal nodes are sufficient to completely describe an S-tree.

As stated earlier, each S-tree represents a Boolean function f. If you have an S-tree and values for the variables x_{1}, x_{2}, ..., x_{n}, then it is quite simple to find out what f(x_{1}, x_{2}, ..., x_{n}) is: start with the root. Now repeat the following: if the node you are at is labelled with a variable x_{i}, then depending on whether the value of the variable is 1 or 0, you go its right or left child, respectively. Once you reach a terminal node, its label gives the value of the function.

Figure 1: S-trees for the function x_{1} ∧ (x_{2} ∨ x_{3})

On the picture, two S-trees representing the same Boolean function, f(x_{1}, x_{2}, x_{3}) = x_{1} ∧ (x_{2} ∨ x_{3}), are shown. For the left tree, the variable ordering is x_{1}, x_{2}, x_{3}, and for the right tree it is x_{3}, x_{1}, x_{2}.

The values of the variables x_{1}, x_{2}, ..., x_{n}, are given as a Variable Values Assignment (VVA)

(x_{1} = b_{1}, x_{2} = b_{2}, ..., x_{n} = b_{n})

with b_{1}, b_{2}, ..., b_{n} ∈ {0,1}. For instance, (x_{1} = 1, x_{2} = 1 x_{3} = 0) would be a valid VVA for n = 3, resulting for the sample function above in the value f(1, 1, 0) = 1 ∧ (1 ∨ 0) = 1. The corresponding paths are shown bold in the picture.

Your task is to write a program which takes an S-tree and some VVAs and computes f(x_{1}, x_{2}, ..., x_{n}) as described above.

The input file contains the description of several S-trees with associated VVAs which you have to process. Each description begins with a line containing a single integer n, 1 ≤ n ≤ 7, the depth of the S-tree. This is followed by a line describing the variable ordering of the S-tree. The format of that line is x_{i1} x_{i2} ...x_{in}. (There will be exactly n different space-separated strings). So, for n = 3 and the variable ordering x_{3}, x_{1}, x_{2}, this line would look as follows:

x3 x1 x2

In the next line the distribution of 0's and 1's over the terminal nodes is given. There will be exactly 2n characters (each of which can be 0 or 1), followed by the new-line character. The characters are given in the order in which they appear in the S-tree, the first character corresponds to the leftmost terminal node of the S-tree, the last one to its rightmost terminal node.

The next line contains a single integer m, the number of VVAs, followed by m lines describing them. Each of the m lines contains exactly n characters (each of which can be 0 or 1), followed by a new-line character. Regardless of the variable ordering of the S-tree, the first character always describes the value of x_{1}, the second character describes the value of x_{2}, and so on. So, the line

110

corresponds to the VVA (x_{1} = 1, x_{2} = 1, x_{3} = 0).

The input is terminated by a test case starting with n = 0. This test case should not be processed.

For each S-tree, output the line "S-Tree #j:", where j is the number of the S-tree. Then print a line that contains the value of f(x_{1}, x_{2}, ..., x_{n}) for each of the given m VVAs, where f is the function defined by the S-tree.

Output a blank line after each test case.

3 x1 x2 x3 00000111 4 000 010 111 110 3 x3 x1 x2 00010011 4 000 010 111 110 0

S-Tree #1: 0011 S-Tree #2: 0011

ACM-ICPC > Regionals > Europe > Mid-Central European Regional Contest > MCERC 1999 I번