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time limit per test: 0.25 sec.

memory limit per test: 65536 KB

memory limit per test: 65536 KB

input: standard

output: standard

output: standard

In the theory of compilers and languages *finite state machines*, also known as *finite automata* are widely used. Deterministic finite automation (DFA) is an ordered set <Σ, U, s, T, φ> where Σ is the finite set called *input alphabet*, U is the finite set of *states*, s from U is the *initial state*, T subset of U is the set of *terminal states* and φ : U × Σ → U is the *transition function*.

The input of the automation is the string α over Σ. Initially the automation is in state s. Each step it reads the first character c of the input string and changes its state to φ(u, c) where u is the current state. After that the first character of the input string is removed and the step repeats. If when its input string is empty the automation is in the terminal state, it is said that it*accepts* the initial string α, in the other case it *rejects* it.

In some cases to simplify the automation the concept of*nonabsorbing edges* is introduced. That is, in addition to φ the function χ : U × Σ → {0, 1} is introduced and when making a transition from some state u with some character c, the leading character is removed from the input string only if χ(u, c) = 0. If χ(u, c) = 1, the input string is kept intact and next transition is performed with the new state and the same character.

It is said that such automation accepts some string α if after a number of steps it transits to the terminal state and the input string becomes empty.

Your task is given the DFA with nonabsorbing edges to compute the number of strings of the given length N that it accepts.

The input of the automation is the string α over Σ. Initially the automation is in state s. Each step it reads the first character c of the input string and changes its state to φ(u, c) where u is the current state. After that the first character of the input string is removed and the step repeats. If when its input string is empty the automation is in the terminal state, it is said that it

In some cases to simplify the automation the concept of

It is said that such automation accepts some string α if after a number of steps it transits to the terminal state and the input string becomes empty.

Your task is given the DFA with nonabsorbing edges to compute the number of strings of the given length N that it accepts.

The first line of the input file contains Σ — a subset of the English alphabet, several different small letters. Next line contains K = | U | — the number of states of the automation (1 ≤ K ≤ 1000). Let states be numbered from 1 to K. Next line contains S (1 ≤ S ≤ K) — the initial state, followed by L = | T | — the number of terminal states and then L different integer numbers ranging from 1 to K — the numbers of terminal states.

Next K lines contain | Σ | integer numbers each and define φ. Next K lines define χ in a similar way. The last line of the input file contains N (1 ≤ N ≤ 60).

Output the only number — the number of different strings of length N over Σ that the given DFA accepts.

In the given example the two strings accepted by the automation are ``aaa'' and ``abb''.

Input

ab

2

1 1 2

2 1

1 2

0 1

0 0

3

2

1 1 2

2 1

1 2

0 1

0 0

3

Output

2

Author: | Andrew Stankevich |

Resource: | Petrozavodsk Summer Trainings 2003 |

Date: | 2003-08-23 |

Codeforces (c) Copyright 2010-2021 Mike Mirzayanov

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