You are given a $2$D integer array points where points[i] $= [x_i, y_i]$ represents a point on the Cartesian plane, and an integer w.

You need to cover all given points using rectangles such that each rectangle has a width of at most w. A rectangle covers every point $(x, y)$ whose $x$ coordinate satisfies $x_1 \leq x \leq x_2$, provided that $0 \leq x_2 - x_1 \leq$ w and the rectangle extends indefinitely along the vertical axis. Multiple points may share the same rectangle, and rectangles are allowed to overlap.

Return the minimum number of rectangles needed to cover all the points.

Note: Points are considered covered even if they lie on the boundary of a rectangle.

Constraints:

• $1 \leq$ points.length $\leq 10^5$

• points[i].length $= 2$

• $0 \leq x_i =$ points[i][0] $\leq 10^9$

• $0 \leq y_i =$ points[i][1] $\leq 10^9$

• $0 \leq$ w $\leq 10^9$

• All pairs $(x_i, y_i)$ are distinct.

Now, let’s take a moment to make sure you’ve correctly understood the problem. The quiz below helps you check if you’re solving the correct problem:

We have a game for you to play. Rearrange the logical building blocks to develop a clearer understanding of how to solve this problem.

Implement your solution in the following coding playground.