${X(n)}$ , where $n$ is any natural number, in which each digit of $X(n)$ is either $0$ or $1$, by the following rules:

Define a sequence of natural number$1. X(1) = 1$

$2.$ We define $X(n+1)$ as a natural number, which can be obtained by replacing the digits of $X(n)$ with $1$ if the digit is $0$, and with $10$ if the digit is $1$.

For example, $X(1)=1$, $X(2)=10$, $X(3)=101$, $X(4)=10110$ and so on.

$A(n)$ is defined as the number of digits in $X(n)$.

$B(n)$ is defined as the number of times $01$ appears in $X(n)$.

For example, $B(1)=0$, $B(2)=0$, $B(3)=1$, $B(4)=1$, $B(5)=3$ and so on.

What is $A(23)+B(23)$?