A15245 | Long Binary String

There is a binary string $t$ of length $10^{100}$ , and initally all of its bits are $\texttt{0}$ . You are given a binary string $s$ , and perform the following operation some times:

- Select some substring of $t$ , and replace it with its XOR with $s$ . $^\dagger$

After several operations, the string $t$ has exactly two bits $\texttt{1}$ ; that is, there are exactly two distinct indices $p$ and $q$ such that the $p$ -th and $q$ -th bits of $t$ are $\texttt{1}$ , and the rest of the bits are $\texttt{0}$ . Find the lexicographically largest $^\ddagger$ string $t$ satisfying these constraints, or report that no such string exists.

$^\dagger$ Formally, choose an index $i$ such that $0 \leq i \leq 10^{100}-|s|$ . For all $1 \leq j \leq |s|$ , if $s_j = \texttt{1}$ , then toggle $t_{i+j}$ . That is, if $t_{i+j}=\texttt{0}$ , set $t_{i+j}=\texttt{1}$ . Otherwise if $t_{i+j}=\texttt{1}$ , set $t_{i+j}=\texttt{0}$ .

$^\ddagger$ A binary string $a$ is lexicographically larger than a binary string $b$ of the same length if in the first position where $a$ and $b$ differ, the string $a$ has a bit $\texttt{1}$ and the corresponding bit in $b$ is $\texttt{0}$ .

The only line of each test contains a single binary string $s$ ( $1 \leq |s| \leq 35$ ).

If no string $t$ exists as described in the statement, output -1. Otherwise, output the integers $p$ and $q$ ( $1 \leq p < q \leq 10^{100}$ ) such that the $p$ -th and $q$ -th bits of the lexicographically maximal $t$ are $\texttt{1}$ .