Fuel‐Optimal Racing

In this problem, you are given a racing game scenario. A car has an initial fuel amount f, and the track from start to finish is composed of n line segments. Each segment is characterized by its length L (in kilometers) and its slope s. The car is allowed to change its speed instantly at any point along the track. When driving on a road with slope s at speed v (in km/h), the fuel consumption per kilometer is given by

$$\max(0, a\,v + b\,s ),$$

where a and b (with b > 0) are given constants. Note that if the expression a·v+b·s is non‐positive, then no fuel is consumed on that kilometer.

The vehicle cannot exceed a maximum speed of v_max km/h. Your task is to decide how to vary the speed along the track so as to minimize the total travel time, while never using more than the initial fuel amount f (there is no refuelling during the race).

Important details:

• You may choose a different speed for each point on the road, even within the same segment.
• On a segment with negative slope (i.e. downhill), it is possible to avoid fuel consumption entirely by keeping the speed not greater than \(-\frac{b\,s}{a}\). However, going too slowly increases the travel time.
• For segments where the fuel consumption function is always positive (for example, flat or uphill), you must use fuel and the consumption per kilometer is a·v+b·s.

The input guarantees that it is always feasible to finish the race (i.e. even by driving very slowly on the fuel‐consuming segments, the total fuel consumed cannot be lower than a certain positive amount).

Your output should be the minimum total travel time needed to complete the race. Answers within an absolute or relative error of 1e-6 are accepted.

inputFormat

The input is read from standard input and has the following format:

f v_max a b n
L₁ s₁
L₂ s₂
…
Lₙ sₙ

Here,

• f is the initial fuel available (a floating point number).
• v_max is the maximum allowed speed in km/h (a floating point number).
• a and b are parameters in the fuel consumption function (floating point numbers, with b > 0).
• n is the number of segments on the track (an integer).
• Each of the next n lines contains two numbers: L (the length of the segment in kilometers) and s (the slope of the segment).

outputFormat

Output a single number: the minimum total travel time to complete the race. The answer is accepted if its absolute or relative error is no more than 1e-6.

sample

100 100 1 1 3
10 0
20 1
30 -1

1.103412

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