Lucy is a frequent arcade visitor. All machines at the arcade give out tickets to exchange for prizes! Lucy's favorite machine works as follows.

The machine only has two buttons: ``Roll'' and ``Withdraw''. Whenever Lucy presses ``Roll'', the machine increases the counter on the screen by a randomly generated \textbf{real} number between $0$ and $d$. At any moment, she can press ``Withdraw`` to get the number of tickets equal to the counter, which gets reset. In case it's not an integer, the machine generously rounds the counter up before handing out the tickets.

More formally, the machine stores a real number $S$, initially equal to 0. On each ``Roll'' press, the machine generates $\Delta$~--- a random \textbf{real} number picked uniformly from the interval $(0, d)$. Then, $S$ increases by the value of $\Delta$. When the ``Withdraw'' button is pressed, the machine rounds $S$ up, giving the player $\lceil S \rceil$ tickets, and then resets $S$ back to zero. Lucy can see the value of $S$ on the screen at any moment with as much precision as she wants, and she can use it to decide whether to roll or withdraw.

Lucy has enough arcade tokens to press ``Roll'' $n$ times, and ``Withdraw'' $k$ times. Find a strategy that maximizes the expected number of tickets Lucy can get, and print this maximum expected number.