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,

0
00
000
11
110
1001

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.

Notes

• 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.

Constraints
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.

NOTE: You only need to implement the given function. Do not read input, instead use the arguments to the function. Do not print the output, instead return values as specified. Still have a question? Checkout Sample Codes for more details.