codeforces#P1787H. Codeforces Scoreboard
Codeforces Scoreboard
Description
You are participating in a Codeforces Round with $n$ problems.
You spend exactly one minute to solve each problem, the time it takes to submit a problem can be ignored. You can only solve at most one problem at any time. The contest starts at time $0$, so you can make your first submission at any time $t \ge 1$ minutes. Whenever you submit a problem, it is always accepted.
The scoring of the $i$-th problem can be represented by three integers $k_i$, $b_i$, and $a_i$. If you solve it at time $t$ minutes, you get $\max(b_i - k_i \cdot t,a_i)$ points.
Your task is to choose an order to solve all these $n$ problems to get the maximum possible score. You can assume the contest is long enough to solve all problems.
Each test contains multiple test cases. The first line contains an integer $t$ ($1 \le t \le 10^4$) — the number of test cases.
The first line of each test case contains one integer $n$ ($1 \le n \le 2 \cdot 10^5$) — the number of problems.
The $n$ lines follow, the $i$-th of them contains three integers $k_i$, $b_i$, $a_i$ ($1\le k_i,b_i,a_i\le 10^9$; $a_i < b_i$), denoting that you get the score of $\max(b_i - k_i \cdot t,a_i)$ if you solve the $i$-th task at time $t$ minutes.
It's guaranteed that the sum of $n$ does not exceed $2 \cdot 10^5$.
For each test case, print a line containing a single integer — the maximum score you can get.
Input
Each test contains multiple test cases. The first line contains an integer $t$ ($1 \le t \le 10^4$) — the number of test cases.
The first line of each test case contains one integer $n$ ($1 \le n \le 2 \cdot 10^5$) — the number of problems.
The $n$ lines follow, the $i$-th of them contains three integers $k_i$, $b_i$, $a_i$ ($1\le k_i,b_i,a_i\le 10^9$; $a_i < b_i$), denoting that you get the score of $\max(b_i - k_i \cdot t,a_i)$ if you solve the $i$-th task at time $t$ minutes.
It's guaranteed that the sum of $n$ does not exceed $2 \cdot 10^5$.
Output
For each test case, print a line containing a single integer — the maximum score you can get.
4
4
10000 1000000000 2006
10000 1000000000 9999
2 999991010 1010
1000000000 1000000000 999999999
6
1 8 1
9 29 4
2 14 3
4 13 1
2 19 5
10 12 5
8
4 10 1
4 19 8
1 14 3
4 15 6
2 9 6
1 11 10
2 19 12
4 19 14
10
5 12 7
5 39 12
2 39 11
3 23 15
5 30 11
3 17 13
5 29 14
3 17 11
3 36 18
3 9 8
3999961003
53
78
180
Note
In the second test case, the points for all problems at each minute are listed below.
| Time | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ |
| Problem $1$ | $7$ | $6$ | $5$ | $\color{red}{4}$ | $3$ | $2$ |
| Problem $2$ | $\color{red}{20}$ | $11$ | $4$ | $4$ | $4$ | $4$ |
| Problem $3$ | $12$ | $10$ | $\color{red}{8}$ | $6$ | $4$ | $3$ |
| Problem $4$ | $9$ | $5$ | $1$ | $1$ | $\color{red}{1}$ | $1$ |
| Problem $5$ | $17$ | $\color{red}{15}$ | $13$ | $11$ | $9$ | $7$ |
| Problem $6$ | $5$ | $5$ | $5$ | $5$ | $5$ | $\color{red}{5}$ |
The points displayed in red denote one of the optimal orders with the score $53$.