Time limit per test: 2 seconds Memory limit: 64 megabytes
input: standard output: standard
In country there was a championship of the game 'What? Where? When?'. The game has the following rules.
There
is a round table divided into 13 equal sectors. The table has an arrow
which can point at a sector. The sectors are numbered from 1 to 13 in
counter-clockwise order. At the beginning of the game, each sector
contains one envelope. Each of the envelopes in sectors 1 through 12
contains one question. The envelope in sector 13 is empty.
There
are two teams, the club team and the TV viewers team. The viewers
submit the questions, which the club team has to answer. These
questions are put into the envelopes on the table. There are also
questions submitted through Internet.
The game consists of
rounds. Each round the current envelope is chosen in the following way:
the arrow is rotated at a random angle (therefore the probability of it
pointing at a particular sector is ).
Then it is rotated in counter-clockwise order until it points at a
sector having a envelope; this envelope is chosen as current. (If after
the random rotation the arrow points at a sector having a envelope, it
is not rotated the second time).
After the current envelope is
chosen, it is taken off the table. If this envelope contains a
question, this question is read out. If the envelope was at sector 13,
the question to be read out is chosen randomly from the questions
submitted through Internet. If the club team answers the question, it
receives a point, otherwise the point comes to the other team. The game
ends when some team has 6 points.
You are given the order of the
envelopes and the probability that the club's team gives the correct
answer to the question in each envelope and to the questions submitted
through Internet. You have to determine the probability of each final
score.
Input
The first line of the input contains single integer (),
which is the number of the questions submitted through Internet. The
next 12 numbers are the probabilities of the club team answering the
question in sectors 1 through 12. The next
numbers are the probabilities of the club team answering the questions
submitted through Internet. All the probabilities are between 0 and 1
and have at most three digits after the decimal point.
Output
The
final score is represented by two numbers separated by a colon. The
first number is the of the club team's points, the second is the
viewers team's points. You have to output the 12 numbers with at least
3 digits after the decimal point, being the probabilities of the scores
'6:0', '6:1', '6:2', '6:3', '6:4', '6:5', '5:6', '4:6', '3:6', '2:6',
'1:6', '0:6' respectively.