# Markov Trains

Time Limit: 1 Second Memory Limit: 32768 KB

Business is not going well for the Dutch Railway Company NS. Due to technical
problems, they are forced to cancel many train services without advance notice.
This is, of course, extremely frustrating for students who travel from home
to school by train.

The worst thing about the whole situation is the randomness of the cancellations.
Nobody knows in advance whether a train service will be cancelled; a cancellation
is not announced until the official departure time. Since there is usually more
than one possible route from home to school, people are often left with an 'if
I had known this in advance I would have taken the other route' sort of feeling.

Recently, the statistics department of the NS found a revolutionary solution
to this problem. They noticed that some train services are cancelled more often
than others. In order to help the passengers, they decided to publish this information.
The new timetables will state not just the time of departure and arrival of
each service, but also its probability of cancellation.

The travel-planner software from the NS, which normally finds the fastest route
between stations, must be updated to find the route which gives the best chance
of arriving in time. This helps passengers to avoid trains that are likely to
cause problems, instead taking a slightly longer, but more reliable route to
school.

Given the new timetables, a departure station and time, a destination station
and a desired arrival time, find the route which gives the best chance of arriving
at the destination in time.

A route in this case is simply an ordered list of stations visited by the passenger,
starting with the departure station and ending with the destination. The passenger
will stick to the route, each time taking the first possible train to the next
station. If a train is cancelled, he will just wait for the next train to that
station.

The chance of arriving in time is taken to be the probability that the passenger,
when following the route as described above, arrives at the destination station
before or at the desired arrival time. When calculating this probability, we
assume that train services are cancelled independently of each other and according
to the probabilities stated in the timetable.

## Input

The first line of the input contains a single positive integer indicating the
number of runs. For each run, the input is as follows:

> A line with a single positive integer n, the number of trains in the timetable
(n <= 100).

> n lines describing the timetable. Each line describes one train, stating
its departure station x, the time of departure tx , its destination station
y (x != y), the time of arrival ty (tx < ty) and its probability of cancellation
p.

Stations are identified by capital letters in the range 'A' ... 'L'. Times are
in the format hh:mm with 00 <= hh < 24 and 00 <= mm < 60. The probability
p is a decimal real number with 0.0 <= p < 1.0. Input elements are separated
by spaces.

> A line with a departure station a, earliest departure time ta , destination
station b (a != b) and desired arrival time tb (ta < tb ). Station identifiers
and times are like those in the timetable.

## Output

The output consists of two lines for each run. The first line of each run contains
the best possible route for the passenger as a list of station identifiers separated
by spaces. The second line contains the probability that the passenger, when
following the given route, arrives on time.

The probability must be formatted as a decimal real number with exactly one
digit before the decimal point, and exactly 4 digits after. The usual rules
for rounding apply: round up if the next digit would be >= 5, otherwise round
down.

Notes

> When changing trains at an intermediate station, the earliest possible
departure time is one minute after the time of arrival.

> All times are on the same day; the journey does not cross midnight.

> It never happens that two or more trains depart from the same station at
the same time to the same destination station.

> The input is such that there is a unique route with maximum probability.

> The passenger will stick to his route, always taking the first available
train to the next station. If a train is cancelled he will wait for the next
train to that station. He will never try to be smart by taking faster trains
or different routes.

## Sample Input

2 3 A 12:00 B 12:15 0.1 A 12:10 B 12:14 0.23 A 12:20 B 12:30 0.456 A 12:00 B 12:30 4 A 12:00 B 12:15 0.1 A 12:05 B 12:13 0.15 B 12:20 C 12:35 0.12 A 12:15 C 12:33 0.4 A 12:00 C 13:00

## Sample Output

A B 0.9895 A B C 0.8668Submit

Source: Northwestern Europe 2002