P9025 [CCC 2021 S3] Lunch Concert

There are $N$ people. The $i$-th person has speed $W_i$ **seconds per meter** and hearing range $D_i$, meaning they can hear music within a distance of at most $D_i$ meters from them. They initially stand at position $P_i$. You are going to hold a concert at position $c$, where $c$ is chosen by you and must be an integer. All $N$ people will move toward you until they can hear you. You want to minimize the sum of the time spent moving by everyone.

The first line contains $N$. The next $N$ lines each contain $P_i, W_i, D_i$ in order.

Output one integer: the minimum possible sum of the time spent moving by everyone. (Note: the answer may exceed $2^{32}$.)

$$1\leq N\leq 200000,0\leq P_i\leq 10^9,1\leq W_i\leq 1000,0\leq D_i\leq 10^9$$ Translated from [CCC2021 S3](https://cemc.math.uwaterloo.ca/contests/computing/past_ccc_contests/2021/ccc/seniorEF.pdf). ###### 2023.8.10 Added a set of hack testdata. Translated by ChatGPT 5