Kotori is practicing making fireworks for the upcoming hanabi taikai1. It takes her nnn minutes to make a single firework, and as she is not really proficient in making fireworks, each firework only has a probability of p×104p times 10^{-4}p×104 to be perfect. After she finishes making a firework, she can just start making the next firework, or take mmm minutes to light all the remaining fireworks finished before. If there is at least one perfect firework among the lit ones, she will be happy and go to rest. Otherwise, she will continue practicing. Can you tell her the minimum expected practicing time before she goes to rest if she takes the optimal strategy?Notice that no matter how many fireworks remain, it always takes mmm minutes to light them all. 1Hanabi taikai: Romaji of the Japanese word "花火大會", which means the firework... err... party?
Kotori is practicing making fireworks for the upcoming hanabi taikai1. It takes her nnn minutes to make a single firework, and as she is not really proficient in making fireworks, each firework only has a probability of p×10−4p times 10^{-4}p×10−4 to be perfect. After she finishes making a firework, she can just start making the next firework, or take mmm minutes to light all the remaining fireworks finished before. If there is at least one perfect firework among the lit ones, she will be happy and go to rest. Otherwise, she will continue practicing. Can you tell her the minimum expected practicing time before she goes to rest if she takes the optimal strategy? Notice that no matter how many fireworks remain, it always takes mmm minutes to light them all. 1Hanabi taikai: Romaji of the Japanese word "花火大會", which means the firework... err... party?
