Problem Statement

You are given an integer N. You can perform the following two types of operations:

• Pay X yen to replace N with $\displaystyle\left\lfloor\frac{N}{A}\right\rfloor$.
• Pay Y yen to roll a die (dice) that shows an integer between 1 and 6, inclusive, with equal probability. Let b be the outcome of the die, and replace N with $\displaystyle\left\lfloor\frac{N}{b}\right\rfloor$.

Here, $\lfloor s \rfloor$ denotes the greatest integer less than or equal to s. For example$, \lfloor 3 \rfloor=3$ and $\lfloor 2.5 \rfloor=2$.
Determine the minimum expected cost paid before N becomes 0 when optimally choosing operations.
The outcome of the die in each operation is independent of other rolls, and the choice of operation can be made after observing the results of the previous operations.

Input

The input is given from Standard Input in the following format:
N A X Y

Output

Print the answer.
Your output will be considered correct if the absolute or relative error from the true answer is at most 10^{-6}.

Constraints

• $1 \leq N \leq 10^{18}$
• $2 \leq A \leq 6$
• $1 \leq X, Y \leq 10^9$
• All input values are integers.

Sample Input 1

3 2 10 20

Sample Output 1

20.000000000000000

The available operations are as follows:

• Pay 10 yen. Replace N with $\displaystyle\left\lfloor\frac{N}{2}\right\rfloor$.
• Pay 20 yen. Roll a die. Let b be the outcome, and replace N with $\displaystyle\left\lfloor\frac{N}{b}\right\rfloor$.

The optimal strategy is to perform the first operation twice.

Sample Input 2

3 2 20 20

Sample Output 2

32.000000000000000

The available operations are as follows:

• Pay 20 yen. Replace N with $\displaystyle\left\lfloor\frac{N}{2}\right\rfloor$.
• Pay 20 yen. Roll a die. Let b be the outcome, and replace N with $\displaystyle\left\lfloor\frac{N}{b}\right\rfloor$.

The optimal strategy is as follows:

• First, perform the second operation to roll the die.
• If the outcome is 4 or greater, then N becomes 0.
• If the outcome is 2 or 3, then N becomes 1. Now, perform the first operation to make N = 0.
• If the outcome is 1, restart from the beginning.

Sample Input 3

314159265358979323 4 223606797 173205080

Sample Output 3

6418410657.7408381

见题面 Input 部分

见题面 Output 部分