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 $$$a\ne b$$$; - 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 $$$n$$$, $$$k$$$ ($$$1 \leq n, k \leq 5\cdot 10^5$$$) — 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".