from math import ceil

def minimumPasses(m, w, p, n):
    min_turns = ceil(n/(m*w))  # min turns without ever buying more workers or machines
    if p > n:
        return min_turns
    turn = 0
    candies = 0
    while candies < n:
        # skip to next turn with buying opportunity
        skip_turns = max([ceil((p - candies) / (m*w)), 1])
        turn += skip_turns
        if turn > min_turns:
            # no reason to continue buying the best strategy is to just save from last min_turn calc
            return min_turns
        candies += m*w * skip_turns
        if candies >= p:
            # get new machines
            new_machines = candies // p
            candies -= new_machines * p
            diff = m-w
            # try to balance workers and machines
            if diff > 0:
                w_buy = min(new_machines, diff)
                w += w_buy
                new_machines -= w_buy
            else:
                m_buy = min(new_machines, abs(diff))
                m += m_buy
                new_machines -= m_buy
            # split remainder after balancing and buy half workers and half machines
            w_buy = new_machines // 2
            w += w_buy
            m += new_machines - w_buy
        # check how long till goal without buying anything
        min_turns = min(ceil((n-candies)/(m*w))+turn, min_turns)
    return turn