Unlucky

Time limit1sMemory limit128 MB

Summary

Find how much water a traveler can deliver to an oasis s meters away, given w total water and carry limit k.

Level: Hard8 of 10
Topics: Math, Greedy
Solved: No attempts yet

Problem

Byteasar is stranded in the middle of a desert, $s$ meters from the nearest oasis.

Luckily he has a compass and knows exactly which direction leads to the oasis. He carries $w$ milliliters of water, but he can hold at most $k$ milliliters at any one time. Walking one meter costs exactly one milliliter of water, and he drinks at a steady rate: for example, after walking $1/3$ of a meter he has drunk $1/3$ of a milliliter. At any moment he may put aside any amount of water to drink later, and he may walk back to collect water he left behind. Sometimes going back for cached water pays off.

Byteasar wants to bring as much water as possible to the oasis. Assume he can always reach the oasis if he drinks wisely. How much water can he bring?

Input

A single line contains three integers $s$ , $w$ , and $k$ ( $10 \le s, w, k \le 10^8$ ): the distance to the oasis, the amount of water he has, and the maximum amount he can carry at once.

Output

Print one real number: the maximum volume of water (in milliliters) that Byteasar can bring to the oasis, printed with exactly three digits after the decimal point.

Examples4

Example 1
Input
```
10 30 10
```
Expected output
```
5.333
```
Example 2
Input
```
30 50 100
```
Expected output
```
20.000
```
Example 3
Input
```
10 100 10
```
Expected output
```
13.998
```
Example 4
Input
```
10 10 10
```
Expected output
```
0.000
```