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F - Fishing

Score : $500$ points

Problem Statement

On a number line, there are $N$ fish swimming.

Fish $i$, which has a weight of $W_i$, is at the coordinate $X_i$ at time $0$ and moves at a speed of $V_i$ in the positive direction.

Takahashi will choose an arbitrary real number $t$ greater than or equal to $0$ and do the following action at time $t$ just once.
Action: Choose an arbitrary real number $x$. Catch all fish whose coordinates are between $x$ and $x+A$, inclusive.

Find the maximum total weight of fish that he can catch.

Constraints

• $1 \leq N \leq 2000$
• $1 \leq A \leq 10^4$
• $1 \leq W_i\leq 10^4$
• $0 \leq X_i\leq 10^4$
• $1 \leq V_i\leq 10^4$
• All values in the input are integers.

Input

The input is given from Standard Input in the following format:

$N$ $A$

$W_1$ $X_1$ $V_1$

$W_2$ $X_2$ $V_2$

$\vdots$

$W_N$ $X_N$ $V_N$

Output

Print the answer.

3 10
100 0 100
1 10 30
10 20 10

111

At time $0.25$, fish $1$, $2$, and $3$ are at the coordinates $25$, $17.5$, and $22.5$, respectively. Thus, the action done at this time with $x=16$ catches all the fish.

3 10
100 100 100
1 10 30
10 20 10

100

One optimal choice is to do the action at time $0$ with $x=100$.

4 10
1000 100 10
100 99 1
10 0 100
1 1 1

1110

One optimal choice is to do the action at time $1$ with $x=100$.