Consider two infinite horizontal lines A and B,
separated ℓ units apart.
The line A has m points at the abscissae a_{1}, …, a_{m}.
The line B has n points at the abscissae b_{1}, …, b_{n}.
Given p different indices i_{1}, …, i_{p} choosen from {1 … m},
and p different indices j_{1}, …, j_{p} choosen from {1 … n},
define d_{k} as the Euclidean distance between a_{ik} and b_{jk},
that is,

d_{k} = | √ |
| . |

You are given ℓ, p, and the points in A and in B.
Pick i_{1}, …, i_{p} and j_{1}, …, j_{p} in order to

minimize max_{k=1..p} d_{k}

Input

Input consists of several cases,
each one with only integer numbers.
Every case begins with four strictly positive numbers ℓ, p, m and n.
Follow a_{1} ≤ a_{2} ≤ … ≤ a_{m−1} ≤ a_{m}.
Follow b_{1} ≤ b_{2} ≤ … ≤ b_{n−1} ≤ b_{n}.
Assume ℓ ≤ 10^{6},
p ≤ min(m, n),
and that the absolute value of each abscissa is at most 10^{6}.

Additionally, assume that m and n are at most 10^{4}.

Output

For every case, print the result with four digits after the decimal point. If you use the long double type, the input cases have no precision issues.

Public test cases

**Input**

1 1 2 2 5 10 9 20 1 2 2 2 5 10 9 20 1000000 4 5 4 300000 300000 300000 300000 300000 -500000 -500000 -500000 -500000 3 2 7 4 0 2 4 6 8 10 12 1 4 7 10

**Output**

1.4142 10.0499 1280624.8475 3.0000

Information

- Author
- Salvador Roura
- Language
- English
- Other languages
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- Official solutions
- C++
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- C++