Minimum Segments

Given an array consisting of a number of intervals, each interval is of type $($a$[$i$],$ b$[$i$]).$ Also provided is an integer k$.$ You are to add exactly one segment $($at least, at most$)$ into the array such that the length of the segment is exactly k and the modified array can be separated into the minimum number of connected sets.

A set of segments is connected if every point in the segment $($from the minimum a$[$i$]$ to the maximum b$[$i$]$ among all i in the set$)$ is covered by some segment in the set.

Example:

The set $[(1,2),(2,3),(1,5)]$ is connected while the set $[(0,2),(3,4)]$ is not because point $2\mathrm{.}5$ is not covered by any segment.

Consider the array a $=[(1,5),(2,4),(6,6),(7,14),(16,19)]$ and given k $=2$; Adding a segment $(5,7)$ into the array, after adding this segment, separate the array into $2$ connected sets:

$[(1,5),(2,4),(5,7),(6,6),(7,14)]$

$[(16,19)]$

However, if we add a segment $(14,16)$ into the array, then we have to separate the array using $3$ connected sets:

$[(1,5),(2,4)]$

$[(6,6),(7,14),(14,16)]$

$[(16,19)]$

So$,2$ connected sets is the minimum answer that we can achieve.

Function Description:

Complete the function minimumDivision in the editor below. minimumDivision has the following parameters:

an integer array a of first parameters of intervals

an integer array b of second parameters of intervals

A: an integer denoting the maximum range of the segment that can be added.

Returns:

an integer denoting the minimum number of sets needed to separate the array after adding the segment.

Constraints:

$1<=$ n $<=2*10^5$

$1<=$ a$[$i$],$ b$[$i$]<=10^9$

$1<=$ k $<=10^9$