You are given two strings s and target, each of length n, consisting of lowercase English letters.

Return the lexicographically smallest string that is both a palindromic permutation of s and strictly greater than target. If no such permutation exists, return an empty string.

Example 1:

Input: s = "baba", target = "abba"

Output: "baab"

Explanation:

The palindromic permutations of s (in lexicographical order) are "abba" and "baab".
The lexicographically smallest permutation that is strictly greater than target is "baab".

Example 2:

Input: s = "baba", target = "bbaa"

Output: ""

Explanation:

The palindromic permutations of s (in lexicographical order) are "abba" and "baab".
None of them is lexicographically strictly greater than target. Therefore, the answer is "".

Example 3:

Input: s = "abc", target = "abb"

Output: ""

Explanation:

s has no palindromic permutations. Therefore, the answer is "".

Example 4:

Input: s = "aac", target = "abb"

Output: "aca"

Explanation:

The only palindromic permutation of s is "aca".
"aca" is strictly greater than target. Therefore, the answer is "aca".

Constraints:

1 <= n == s.length == target.length <= 300
s and target consist of only lowercase English letters.