def lily_internal(a: list[int], reverse: bool=True):
    targets = {n: i for i, n in enumerate(sorted(a, reverse=reverse))}
    swaps = 0
    for i, n in enumerate(a):
        while i != targets[n]:
            # Swap the current value into the correct place.
            a[targets[n]], a[i] = n, a[targets[n]]
            swaps += 1
            n = a[i]
    return swaps

def lilysHomework(a: list[int]):
    # lily_internal() is destructive, so send a copy the first time.
    return min(lily_internal(list(a), False), lily_internal(a, True))