from typing import Protocol


class Comparable(Protocol):
    def __lt__(self, other: object, /) -> bool: ...


def adaptive_merge_sort[T: Comparable](sequence: list[T]) -> list[T]:
    if len(sequence) < 2:
        return sequence

    aux = sequence[:]
    adaptive_merge_sort_helper(sequence, aux, 0, len(sequence) - 1)
    return sequence


def adaptive_merge_sort_helper[T: Comparable](
    array: list[T], aux: list[T], low: int, high: int
) -> None:
    if high <= low:
        return

    mid = (low + high) // 2

    adaptive_merge_sort_helper(aux, array, low, mid)
    adaptive_merge_sort_helper(aux, array, mid + 1, high)

    if not array[mid + 1] < array[mid]:
        array[low : high + 1] = aux[low : high + 1]
        return

    merge(array, aux, low, mid, high)


def merge[T: Comparable](
    array: list[T], aux: list[T], low: int, mid: int, high: int
) -> None:
    i, j = low, mid + 1

    for k in range(low, high + 1):
        if i > mid or j > high:
            if i > mid:
                aux[k] = array[j]
                j += 1
            else:
                aux[k] = array[i]
                i += 1
        elif not array[j] < array[i]:
            aux[k] = array[i]
            i += 1
        else:
            aux[k] = array[j]
            j += 1

    for k in range(low, high + 1):
        array[k] = aux[k]