Binary Love

Alice and Bob are playing a game on a binary string $S$ of length $N$. Alice wants to make the number of substrings ``01`` and ``10`` equal, and both counts must be non-zero. Formally, let $c_{01}$ denote the number of indices $i$ ($1 \le i \lt N$) such that $S_i = 0$ and $S_{i+1} = 1$, and similarly let $c_{10}$ denote the number of indices $i$ ($1 \le i \lt N$) such that $S_i = 1$ and $S_{i+