16B. Smallest Mex String

You are given a binary string $s$.

You have to split $s$ into one or more non-empty contiguous substrings. Each character of $s$ must belong to exactly one substring.

For a binary string $x$, define $mex(x)$ as the smallest non-negative integer that does not appear in $x$ as a digit. For example, $mex(\texttt{1})=0$, $mex(\texttt{0})=1$, and $mex(\texttt{01})=2$.

After choosing a split, write the values of $mex$ of the substrings from left to right, without spaces. This gives a string of digits.

Find the lexicographically smallest string that can be obtained.

We use the usual definition of lexicographic order: if one string is a proper prefix of the other, the shorter string is smaller.