Binary Period

Let's say string $$$s$$$ has period $$$k$$$ if $$$s_i = s_{i + k}$$$ for all $$$i$$$ from $$$1$$$ to $$$|s| - k$$$ ($$$|s|$$$ means length of string $$$s$$$) and $$$k$$$ is the minimum positive integer with this property.

Some examples of a period: for $$$s$$$="0101" the period is $$$k=2$$$, for $$$s$$$="0000" the period is $$$k=1$$$, for $$$s$$$="010" the period is $$$k=2$$$, for $$$s$$$="0011" the period is $$$k=4$$$.

You are given string $$$t$$$ consisting only of 0's and 1's and you need to find such string $$$s$$$ that:

1. String $$$s$$$ consists only of 0's and 1's;
2. The length of $$$s$$$ doesn't exceed $$$2 \cdot |t|$$$;
3. String $$$t$$$ is a subsequence of string $$$s$$$;
4. String $$$s$$$ has smallest possible period among all strings that meet conditions 1—3.

Let us recall that $$$t$$$ is a subsequence of $$$s$$$ if $$$t$$$ can be derived from $$$s$$$ by deleting zero or more elements (any) without changing the order of the remaining elements. For example, $$$t$$$="011" is a subsequence of $$$s$$$="10101".