Implement a quantum oracle on $N$ qubits which checks whether the bits in the input vector $\vec{x}$ form a periodic bit string (i.e., implements the function $f(\vec{x}) = 1$ if $\vec{x}$ is periodic, and 0 otherwise).

A bit string of length $N$ is considered periodic with period $P$ ($1 \le P \le N - 1$) if for all $i \in [0, N - P - 1]$ $x_i = x_{i + P}$. Note that $P$ does not have to divide $N$ evenly; for example, bit string "01010" is periodic with period $P = 2$.

You have to implement an operation which takes the following inputs:

• an array of $N$ ($2 \le N \le 7$) qubits $x$ in an arbitrary state (input register),
• a qubit $y$ in an arbitrary state (output qubit),

and performs a transformation $|x\rangle|y\rangle \rightarrow |x\rangle|y \oplus f(x)\rangle$. The operation doesn't have an output; its "output" is the state in which it leaves the qubits. Note that the input register $x$ has to remain unchanged after applying the operation.

Your code should have the following signature:

namespace Solution {
    open Microsoft.Quantum.Primitive;
    open Microsoft.Quantum.Canon;
operation Solve (x : Qubit[], y : Qubit) : Unit {
    body (...) {
        // your code here
    }
    adjoint auto;
}

}

Note: the operation has to have an adjoint specified for it; adjoint auto means that the adjoint will be generated automatically. For details on adjoint, see Operation Definitions.

You are not allowed to use measurements in your operation.