class Hashing:
    __slots__ = ["mod", "h", "p"]

    def __init__(self, s: str, base: int, mod: int):
        self.mod = mod
        self.h = [0] * (len(s) + 1)
        self.p = [1] * (len(s) + 1)
        for i in range(1, len(s) + 1):
            self.h[i] = (self.h[i - 1] * base + ord(s[i - 1])) % mod
            self.p[i] = (self.p[i - 1] * base) % mod

    def query(self, l: int, r: int) -> int:
        return (self.h[r] - self.h[l - 1] * self.p[r - l + 1]) % self.mod


class Solution:
    def minimumTimeToInitialState(self, word: str, k: int) -> int:
        hashing = Hashing(word, 13331, 998244353)
        n = len(word)
        for i in range(k, n, k):
            if hashing.query(1, n - i) == hashing.query(i + 1, n):
                return i // k
        return (n + k - 1) // k