"""Manacher's algorithm: longest palindromic substring in O(n)."""
from __future__ import annotations


def longest_palindrome(s: str) -> str:
    if s == "":
        return ""
    # Transform: insert sentinels so every palindrome has odd length.
    t = "^#" + "#".join(s) + "#$"
    n = len(t)
    p = [0] * n
    c = r = 0
    for i in range(1, n - 1):
        mirror = 2 * c - i
        if i < r:
            p[i] = min(r - i, p[mirror])
        # expand
        while t[i + p[i] + 1] == t[i - p[i] - 1]:
            p[i] += 1
        if i + p[i] > r:
            c = i
            r = i + p[i]
    # find center with max radius
    center = 1
    for i in range(1, n - 1):
        if p[i] > p[center]:
            center = i
    start = (center - p[center]) // 2
    return s[start:start + p[center]]


def all_palindromic_substrings(s: str) -> list[str]:
    """Return all distinct palindromic substrings (may be large)."""
    res = set()
    n = len(s)
    for i in range(n):
        # odd
        l = r = i
        while l >= 0 and r < n and s[l] == s[r]:
            res.add(s[l:r + 1])
            l -= 1
            r += 1
        # even
        l, r = i, i + 1
        while l >= 0 and r < n and s[l] == s[r]:
            res.add(s[l:r + 1])
            l -= 1
            r += 1
    return sorted(res)