Covering with intervals P76554

Statement

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Given a natural k and several numbers x₁, …, x_n, we want to find the smallest possible set of closed intervals of length k that cover those numbers. In other words, we must find a set of intervals {[y₁, y₁ + k], …, [y_m, y_m + k]} such that

for every x_i, there exists some j such that x_i ∈ [y_j, y_j + k];
m is minimum.

For instance, if k=10 and the x_i’s are 14, 19, 23 and 27, a possible solution is {[12, 22], [1.8, 2.8]}, since every x_i belongs to (at least) one of the two intervals, and it is not possible to cover the four numbers with a single interval.

Input

Input consists of several cases, each of which starts with k, followed by n, followed by n different numbers. All numbers in the input are integers. Assume 1 ≤ k, n ≤ 10⁵.

Output

For every case, print the minimum number of closed intervals of length k that cover the given numbers.

Public test cases

Input

10  4  14 19 23 27
100 6  175 350 50 300 150 20
10  2  -25 -35

Output

2
3
1

Information

Author: Enric Rodriguez
Language: English
Translator: Enric Rodriguez
Original language: Catalan
Other languages: Catalan
Official solutions: C++
User solutions: C++