The Binary Dragon Coil

The 'Dragon Coil' sequence $S_n$ is defined recursively: $S_0 = "0"$. For $n > 0$, $S_n = S_{n-1} + "1" + \text{reverse}(\text{flip}(S_{n-1}))$, where 'flip' changes '0' to '1' and '1' to '0'. For example, $S_1 = "011"$ and $S_2 = "0111001"$. Given $N$ and $K$, find the $K$-th character (0-indexed) of $S_N$. Constraints: $N \le 60$, $K < 2^{N+1}-1$.