You are given a $2$D integer array tires where tires[i] = [f_i, r_i] indicates that the $i$th tire takes $f_i$ seconds to complete its first lap, and each subsequent consecutive lap completed using the same tire takes $r_i$ times the time of the previous lap.

Specifically, the $x$th consecutive lap run on the $i$th tire takes $f_i imes r_i^{(x - 1)}$ seconds.

You are also given an integer changeTime and an integer numLaps.

The race consists of numLaps laps. You may begin the race using any tire for free. You can change to any tire (including a brand new tire of the same type) at any time, but doing so adds changeTime seconds to your total time. There is no limit to the number of tires available of each type.

Return the minimum total time required to finish all numLaps laps.

Constraints:

• $1 \leq$ tires.length $\leq 10^5$

• tires[i].length $= 2$

• $1 \leq f_i$, changeTime $\leq 10^5$

• $2 \leq r_i \leq 10^5$

• $1 \leq$ numLaps $\leq 1000$

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.