write a go solution for Description:
This is the hard version of the problem. The only difference is the constraints on n and k. You can make hacks only if all versions of the problem are solved.

You have a string s, and you can do two types of operations on it:

- Delete the last character of the string.
- Duplicate the string: s:=s+s, where + denotes concatenation.

You can use each operation any number of times (possibly none).

Your task is to find the lexicographically smallest string of length exactly k that can be obtained by doing these operations on string s.

A string a is lexicographically smaller than a string b if and only if one of the following holds:

- a is a prefix of b, but aneb;
- In the first position where a and b differ, the string a has a letter that appears earlier in the alphabet than the corresponding letter in b.

Input Format:
The first line contains two integers n, k (1<=n,k<=5*10^5) — the length of the original string s and the length of the desired string.

The second line contains the string s, consisting of n lowercase English letters.

Output Format:
Print the lexicographically smallest string of length k that can be obtained by doing the operations on string s.

Note:
In the first test, it is optimal to make one duplication: "dbcadabc" to "dbcadabcdbcadabc".

In the second test it is optimal to delete the last 3 characters, then duplicate the string 3 times, then delete the last 3 characters to make the string have length k.

"abcd" to "abc" to "ab" to "a" to "aa" to "aaaa" to "aaaaaaaa" to "aaaaaaa" to "aaaaaa" to "aaaaa".. Output only the code with no comments, explanation, or additional text.