You are given a $2$D integer array intervals, where intervals[i] = [start_i, end_i] represents the start and end points of the $i^{ ext{th}}$ interval. You are also given an integer k.

A collection of intervals can be partitioned into connected groups where any two intervals that overlap or touch at their endpoints belong to the same group.

You can insert one new interval of length k, represented as [x, x + k] for any integer x, into intervals.

Return the minimum number of connected groups obtained after inserting the new interval.

Constraints:

• $1 \leq$ intervals.length $\leq 10^5$

• intervals[i] == [start_i, end_i]

• $1 \leq$ start_i $\leq$ end_i $\leq 10^9$

• $1 \leq$ k $\leq 10^9$

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