Prove that A is non-regular using K-Complexity Non regularity theorem
Suppose that $A$ is regular. Then there exists a constant $c$ such that for all $x \in \Sigma^*$ and for all $n$ such that $Y^A_{x,n}$ exists, $C(Y^A_{x,n}) \leq \log n + c$.

Let us take $x = 0^{2m}1$. Then $Y^A_{x,n}$ exists as long as $n \leq 2^m$ – in fact, $Y^A_{x,n}$ is just the $n$th binary string of length $m$. Therefore
$$
C(\text{$n$th binary string of length $m$}) \leq \log n + c.
$$
In particular, choosing $n=1$, we get
$$
C(0^m) \leq c.
$$
Clearly this fails for large enough $m$. This contradiction shows that $A$ is not regular.