B	Infinite Stable Integer

For any infinite-length decimal integer S = d₁d₂d₃d₄... (0 ≤ d_i ≤ 9, i ≥1), let prefix(S,p) be the integer formed by the first p digits of S (i.e. d₁d₂d₃...d_p), and F(S,i,p) be the percentage of digit i in prefix(S,p). For example, if S = 122312231223..., F(S,2,7) =

We say S is stable if and only if every for digit i (0 ≤ i ≤ 9), there exists a real number L(i) such that

Given three positive integers M, X and Y (0 ≤ X ≤ Y < M), and 10 pairs of integers (A(0), B(0)), (A(1),B(1)), ..., (A(9),B(9)), find an infinite stable integer S such that:

1. Every L(i) satisfies A(i) ≤ L(i) ≤ B(i)

2. For every integer p ≥ 1, X ≤ (prefix(S, p) mod M )≤ Y.

If there are more than one solution, maximize the average value of all the digits in S. Since S is stable, it can be proven that the average value converges.

For example, if M=9, X=1 and Y=8, B(3)=B(4)=100, all other A(i) and B(i) are zero, then the optimal S is 44(4444443)*, where * means "repeated forever". It's not hard to see that prefix(S,p) will never be a multiple of 9, and L(3)=, L(4)=, all other L(i)=0.

Input

There will be multiple test cases. Each test case contains 23 integers: M, X, Y, A(0), A(1), ..., A(9), B(0), B(1), ..., B(9). 2 ≤ M ≤ 50, 0 ≤ X ≤ Y < M, 0 ≤ A(i) ≤ B(i) ≤ 100.

Output

For each test case, print case number and the maximal average value rounded to 8 decimal places. If no infinite stable integer can be found, print "NO SOLUTION" instead. Look at the output for sample input for details.

Sample Input

2 1 1 0 0 0 0 0 0 0 0 0 0 20 20 20 20 20 20 20 20 20 20

9 1 8 0 0 0 0 0 0 0 0 0 0 0 0 0 100 100 0 0 0 0 0

8 0 7 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2

19 2  3  0  0  0  0  0  0  0  0  0  0 100 100 100 100 100 100 100 100 100 100

Output for Sample Input

Case 1: 5.00000000

Case 2: 3.85714286

Case 3: NO SOLUTION

Case 4: 1.00000000

Problemsetter: Bin Jin, Rujia Liu