21406_PitStopStrategy

The speed of a racing car, all other factors being equal, depends on the amount of fuel it carries. In general, the weight of the fuel slows the car. In addition, the weight of the fuel increases fuel consumption. It is therefore an advantage to carry as little fuel as possible. Any amount of fuel may be loaded at the beginning of the race, and refueling pit shops may be made during the race. The time consumed for the pit stop increases with the amount of fuel loaded. Your task is to determine the optimal fueling and pit stop strategy for each of a number of cars, based on measurements taken immediately before the race.

Standard input consists of several lines of input, each containing:

• the number of laps in the race (integer less than or equal to 100)

• the theoretical lap time [second] of the car on an empty tank (float)

• the increase in lap time (float) per liter of fuel carried at the beginning of the lap (float)

• the theoretical fuel consumption [liters per lap] on an empty tank (float)

• the increase in fuel consumption per liter of fuel carried at the beginning of the lap (float; strictly less than 1)

• the time [seconds] for a pit stop taking on no fuel (float)

• the extra pit stop time per liter of fuel loaded (float)

For each input line, print the following information

• one line containing the seven input numbers in order

• one line containing three integers:
  total race time
  the amount of fuel to be loaded initially
  the number of pit stops

• for each pit stop, a line containing:
  the number of laps completed since the start of the race at the time of the pit stop
  the amount of fuel to be taken on

All floating point results should be printed to 6 significant figures. All numbers on a single line be separated by a single space.

If there are several optimal strategies, break the tie with an earlier pit stop.

3 100 0 10 0 20 0
3 100 0 10 .1 20 0
3 100 2 10 0 20 1
3 100 4 10 0 20 1
3 100 2 10 .1 20 1

3 100 0 10 0 20 0
300 30 0
3 100 0 10 0.1 20 0
300 37.1742 0
3 100 2 10 0 20 1
410 20 1
2 10
3 100 4 10 0 20 1
480 10 2
1 10
2 10
3 100 2 10 0.1 20 1
422.469 23.4568 1
2 11.1111