Deterministic Finite Automaton

Deterministic finite automaton(DFA) is a finite state machine that accepts/rejects finite strings of symbols and only produces a unique computation (or run) of the automation for each input string.

DFAs can be represented using state diagrams. For example, in the automaton shown below, there are three states: S0, S1, and S2 (denoted graphically by circles). The automaton takes a finite sequence of 0s and 1s as input. For each state, there is a transition arrow leading out to a next state for both 0 and 1. Upon reading a symbol, a DFA jumps deterministically from a state to another by following the transition arrow. For example, if the automaton is currently in state S0 and current input symbol is 1 then it deterministically jumps to state S1. A DFA has a start state (denoted graphically by an arrow coming in from nowhere) where computations begin, and a set of accept states (denoted graphically by a double circle) which help define when a computation is successful.


These are some strings above DFA accepts,


You are given a DFA in input and an integer N. You have to tell how many distinct strings of length N the given DFA accepts. Return answer modulo 109+7.

- Assume each state has two outgoing edges(one for 0 and one for 1). Both outgoing edges won’t go to the same state.
- There could be multiple accept states, but only one start state.
- A start state could also be an accept state.

Input format
- States are numbered from 0 to K-1, where K is total number of states in DFA.
- You are given three arrays A, B, C and two integers D and N.
- Array A denotes a 0 edge from state numbered i to state A[i], for all 0 ≤ i ≤ K-1
- Array B denotes a 1 edge from state numbered i to state B[i], for all 0 ≤ i ≤ K-1
- Array C contains indices of all accept states.
- Integer D denotes the start state.
- Integer N denotes you have to count how many distinct strings of length N the given DFA accepts.

1 ≤ K ≤ 50
1 ≤ N ≤ 104

Example :

For the DFA shown in image, input is
A = [0, 2, 1]
B = [1, 0, 2]
C = [0]
D = 0

Input 1
N = 2
Strings '00' and '11' are only strings on length 2 which are accepted. So, answer is 2.

Input 2
N = 1
String '0' is the only string. Answer is 1.

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