You are given a $2$D array of integers coordinates of size $n$, where coordinates[i] $= [x_i, y_i]$ represents a point on a $2$D plane, and an integer k.

A sequence of points is called an increasing path if for any two consecutive points $(x_i, y_i)$ and $(x_{i+1}, y_{i+1})$ in the sequence, both $x_i < x_{i+1}$ and $y_i < y_{i+1}$ hold.

Return the maximum length of an increasing path that contains the point coordinates[k].

Constraints:

• $1 \leq n ==$ coordinates.length $\leq 10^5$

• coordinates[i].length $== 2$

• $0 \leq$ coordinates[i][0], coordinates[i][1] $\leq 10^9$

• All elements in coordinates are distinct.

• $0 \leq$ k $\leq n - 1$

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.