You are given a $0$ indexed integer array lights of length n, representing a road with n consecutive positions indexed from $0$ to n $- 1$.

An existing light at index i has an illumination radius of lights[i]. When lights[i] > 0, it illuminates all positions from $\max(0, i -$ lights[i]$)$ to $\min($n $- 1, i +$ lights[i]$)$. If lights[i] = 0, there is no initial light active at index i.

You may install additional lights at any position. Each newly installed light has an illumination radius of $1$, which illuminates all positions from $\max(0, i - 1)$ to $\min($n $- 1, i + 1)$.

Return the minimum number of additional lights needed so that every position from $0$ to n $- 1$ is illuminated.

Constraints:

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

• $0 \leq$ lights[i] $\leq$ n

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.