You are given an integer n representing the number of positions on a line, indexed from $0$ to n $- 1$, and an integer brightness representing the minimum number of positions that must be illuminated.

Placing a light bulb at position p illuminates position p as well as its adjacent positions p $- 1$ and p $+ 1$ (if they exist within the line).

You are also given a $2$D integer array intervals, where intervals[i] = [start_i, end_i] represents an inclusive active time interval $[$ start_i, end_i $]$ of length end_i $-$ start_i $+ 1$.

Each light bulb consumes $1$ unit of energy per unit of time while active. The total active time is the total length of the union of all intervals in intervals.

Return the minimum total energy required to illuminate at least brightness positions across the entire active time.

Constraints:

• $1 \leq$ n $\leq 10^6$

• $1 \leq$ brightness $\leq$ n

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

• intervals[i] == [start_i, end_i]

• $0 \leq$ start_i $\leq$ end_i $\leq 10^9$

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.