Problem H: Ramen Shop

Ron is a master of a ramen shop.

Recently, he has noticed some customers wait for a long time. This has been caused by lack of
seats during lunch time. Customers loses their satisfaction if they waits for a long time, and
even some of them will give up waiting and go away. For this reason, he has decided to increase
seats in his shop. To determine how many seats are appropriate, he has asked you, an excellent
programmer, to write a simulator of customer behavior.

Customers come to his shop in groups, each of which is associated with the following four
parameters:

Ti : when the group comes to the shop

Pi : number of customers

Wi : how long the group can wait for their seats

Ei : how long the group takes for eating

The i-th group comes to the shop with Pi customers together at the time Ti . If Pi successive
seats are available at that time, the group takes their seats immediately. Otherwise, they waits
for such seats being available. When the group fails to take their seats within the time Wi
(inclusive) from their coming and strictly before the closing time, they give up waiting and go
away. In addition, if there are other groups waiting, the new group cannot take their seats until
the earlier groups are taking seats or going away.

The shop has N counters numbered uniquely from 1 to N. The i-th counter has Ci seats. The
group prefers “seats with a greater distance to the nearest group.” Precisely, the group takes
their seats according to the criteria listed below. Here, SL denotes the number of successive
empty seats on the left side of the group after their seating, and SR the number on the right
side. SL and SR are considered to be infinity if there are no other customers on the left side
and on the right side respectively.

Prefers seats maximizing min{SL, SR}.

If there are multiple alternatives meeting the first criterion, prefers seats maximizing
max{SL, SR}.

If there are still multiple alternatives, prefers the counter of the smallest number.

If there are still multiple alternatives, prefers the leftmost seats.

When multiple groups are leaving the shop at the same time and some other group is waiting
for available seats, seat assignment for the waiting group should be made after all the finished
groups leave the shop.

Your task is to calculate the average satisfaction over customers. The satisfaction of a customer
in the i-th group is given as follows:

If the group goes away without eating, -1.

Otherwise, (Wi - ti )/Wi where ti is the actual waiting time for the i-th group (the value ranges between 0 to 1 inclusive).

Input

The input consists of multiple datasets.

Each dataset has the following format:

N M T
C1 C2 ... CN
T1 P1 W1 E1
T2 P2 W2 E2
...
TM PM WM EM

N indicates the number of counters, M indicates the number of groups and T indicates the
closing time. The shop always opens at the time 0. All input values are integers.

You can assume that 1 ≤ N ≤ 100, 1 ≤ M ≤ 10000, 1 ≤ T ≤ 109, 1 ≤ Ci ≤ 100, 0 ≤ T1 < T2 < ... < TM < T, 1 ≤ Pi ≤ max Ci, 1 ≤ Wi ≤ 109 and 1 ≤ Ei ≤ 109.

The input is terminated by a line with three zeros. This is not part of any datasets.

Output

For each dataset, output the average satisfaction over all customers in a line. Each output value
may be printed with an arbitrary number of fractional digits, but may not contain an absolute
error greater than 10-9.

Sample Input

1 4 100
7
10 1 50 50
15 2 50 50
25 1 50 50
35 3 50 50
1 2 100
5
30 3 20 50
40 4 40 50
1 2 100
5
49 3 20 50
60 4 50 30
1 2 100
5
50 3 20 50
60 4 50 30
2 3 100
4 2
10 4 20 20
30 2 20 20
40 4 20 20
0 0 0

Output for the Sample Input

0.7428571428571429
0.4285714285714285
0.5542857142857143
-0.1428571428571428
0.8000000000000000