Traffic Lights Synchronization

Consider a straight road called Main Street with several traffic lights positioned at various locations. Each traffic light repeatedly turns red for r seconds and green for g seconds (r and g may be different for each light). At time 0, every traffic light has just turned red.

An ideal car appears magically at the west end of Main Street at a uniformly random real time in the interval \([0, 2019!]\), and it drives eastward at a constant speed of 1 meter/second. When the car reaches a traffic light located at position x (in meters), it observes the light; if the light is green it continues, but if it is red the car is forced to stop immediately and remains there forever.

Your task is to compute two probabilities:

1. The probability that the car will make it through all the traffic lights without stopping.
2. If the car is forced to stop, the probability that the first red light it encounters is a specific traffic light (when the lights are considered in order along Main Street from west to east).

Note: For a traffic light with parameters r and g, its cycle length is \(T = r+g\). At a time \(t\) when the car reaches a light at position \(x\), the condition for the light to be green is
\( (t+x) \bmod T \in [r, r+g) \).
Because the departure time is uniformly distributed over an interval that is a multiple of each cycle, the probability that a particular light is green when reached is \(\frac{g}{r+g}\), and similarly red is \(\frac{r}{r+g}\). Assuming the lights are processed in order of increasing position, the overall probability of never stopping is the product

[ P(\text{success}) = \prod_{i=1}^{n} \frac{g_i}{r_i+g_i}. ]

Furthermore, the probability that the car is stopped for the first time at the (i)-th light (after having seen all previous lights green) is

[ P(\text{first red at light } i) = \left(\prod_{j=1}^{i-1} \frac{g_j}{r_j+g_j}\right) \cdot \frac{r_i}{r_i+g_i}. ]

Input details follow below.

inputFormat

The input begins with a single integer n (the number of traffic lights). Each of the following n lines contains three space‐separated integers:

• x: the position (in meters) of a traffic light along Main Street,
• r: the duration (in seconds) the light is red,
• g: the duration (in seconds) the light is green.

The traffic lights can be given in any order. You should process them in order of increasing x (i.e. from west to east).

outputFormat

The output consists of two lines:

1. The first line contains a floating point number representing the probability that the car makes it through all the lights without stopping. Print this probability with 6 decimal places.
2. The second line contains n floating point numbers (each printed with 6 decimal places) separated by a space. The i-th number is the probability that the car is forced to stop for the first time at the i-th traffic light (when sorted in increasing order of position).

Note: The probabilities should be computed exactly as follows:

• \(P(\text{success}) = \prod_{i=1}^{n} \frac{g_i}{r_i+g_i}\).
• \(P(\text{first red at light } i) = \left(\prod_{j=1}^{i-1} \frac{g_j}{r_j+g_j}\right) \cdot \frac{r_i}{r_i+g_i}\).

sample

2
10 30 30
50 20 40

0.333333
0.500000 0.166667

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