Barcelona's trams P38716

Quite recently, the City of Barcelona has included trams to its “efficient” public transport. As expected, the result has been a nice set of accidents of outstanding originality and beauty. But diminishing aesthetic reasons, the Mayor of Barcelona has decided to reduce the delay caused by the accidents. After a thorough study the following model has been established.

Every tram must go from an initial point $P_0$ to a final point $P_n$ visiting the intermediate points $P_1$, …, $P_{n-1}$ in this order. For every $1 \le i \le n$, let $S_i$ be the section that goes from $P_{i-1}$ to $P_i$. Every such section must be travelled at uniform speed $v_i$, which is chosen by the driver at $P_{i-1}$. Let $M_i$ be the maximum possible speed of the tram at $S_i$, and assume that the chosen speed is $0 < v_i \le M_i$. Then the probability of crashing in $S_i$ is $v_i/M_i$. When a crash happens, the tram uses an efficient recovery system that lasts only 10 seconds. Afterwards, the tram reaches $P_i$ using an auxiliary (slow but safe) engine, which provides a speed of 5 meters per second and guarantees no more crashes in $S_i$.

For instance, assume that the section length is 300 meters, and that the current maximum speed is 25 meters per second. If the driver chooses to travel at $25 \, m/s$, the tram will crash for sure. Since this can happen anywhere between $P_{i-1}$ and $P_i$, for the sake of computation we can assume that it will take place exactly in the middle point (after 150 meters). Therefore, on the average the tram will spend 6 seconds to reach the middle point, 10 seconds to recover from the crash, and 30 seconds to reach $P_i$, for a total of 46 seconds. By contrast, if the tram starts traveling at $15 \, m/s$, with probability $0.6$ it will crash and spend $10 + 10 + 30 = 50$ seconds, and with probability $0.4$ it will reach $P_i$ after 20 seconds without any crash. The average time in this case is thus just $0.6*50 + 0.4*20 = 38$ seconds.

When the tram reaches every $P_i$, it stops for a few seconds regardless of having crashed in $S_i$ or not; these few seconds (for simplicity, we consider them to be 0) are enough to (almost) repair the tram: the maximum speed reduces by $1 \, m/s$ after every crash. In other words, if we call the initial maximum speed $M_0$, then we have $M_i = M_0 - C_i$, where $0 \le C_i \le i-1$ is the total number of crashes suffered in $S_1$, …, $S_{i-1}$.

Write a program to print the optimal average travel time given the initial maximum speed and the length of every section.

Input

Input consists of several cases, each one with $M_0$ (a real number between 5 and 1000), $n$ (an integer number between 1 and $M_0 - 1$), and the length of every section (each one a real number between 100 and 1000).

@par @ @̧tmp=0 @̧tmp=0 @̣tmp@̣tmp by@̣tmp =-@̧tmp=@̣breite @̣leftskip

Output

For every case, print the optimal average travel time with four digits after the decimal point. The input cases have no precision issues.