Dumb Bones

You are setting up a straight line of dominoes, each standing on end, so that later they can be toppled over for entertainment.

The tricky part of setting up dominoes is that if you make a mistake and knock one over while you are placing it, it also knocks down the entire run of consecutive dominoes immediately adjacent to it on one side, partially ruining your work.

For example, suppose the dominoes already placed form the pattern DD__DxDDD_D, where D is a standing domino, _ is an empty spot, and x is the position where you are about to place a domino. If the domino you place at position x falls, it may knock over the single domino to its left, or the run of three dominoes to its right, forcing you to place those again.

This human error is somewhat unavoidable, but you can make the odds a little more favourable by using a placing technique that makes dominoes fall in one direction more often than the other.

Given the number of dominoes you want to set up, and the probability that any single domino you place falls to the left or to the right, determine the average (expected) number of dominoes you will need to place before you finish, assuming you use an optimal placing strategy.

The input consists of up to 100 test cases. Each test case is a single line containing the number of dominoes to place, $n$ ($1 \le n \le 1000$), followed by two nonnegative real numbers $P_l$ and $P_r$, the probability that a domino falls to the left or to the right when it is placed. You may assume $0 < P_l + P_r \le 0.5$.

The last test case is followed by a line containing a single 0, which is not processed.

For each test case, output on its own line the expected number of dominoes that must be placed before you finish, rounded to exactly two digits after the decimal point.