Problem 1. Bovine Acrobatics

USACO 2023 December Contest, Silver

Farmer John has decided to make his cows do some acrobatics! First, FJ weighs his cows and finds that they have $N$ ($1\le N\le 2\cdot 10^5$) distinct weights. In particular, for each $i\in [1,N]$, $a_i$ of his cows have a weight of $w_i$ ($1\le a_i\le 10^9, 1\le w_i\le 10^9$).

His most popular stunt involves the cows forming

balanced towers

. A

tower

is a sequence of cows where each cow is stacked on top of the next. A tower is

balanced

if every cow with a cow directly above it has weight at least $K$ ($1\le K\le 10^9$) greater than the weight of the cow directly above it. Any cow can be part of at most one balanced tower.

If FJ wants to create at most $M$ ($1 \le M \le 10^9$) balanced towers of cows, at most how many cows can be part of some tower?

INPUT FORMAT (input arrives from the terminal / stdin):

The first line contains three space-separated integers, $N$, $M$, and $K$.

The next $N$ lines contain two space-separated integers, $w_{i}$ and $a_i$. It is guaranteed that all $w_i$ are distinct.

OUTPUT FORMAT (print output to the terminal / stdout):

Output the maximum number of cows in balanced towers if FJ helps the cows form towers optimally.

SAMPLE INPUT:

3 5 2
9 4
7 6
5 5

SAMPLE OUTPUT:

14

FJ can create four balanced towers with cows of weights 5, 7, and 9, and one balanced tower with cows of weights 5 and 7.

SAMPLE INPUT:

3 5 3
5 5
7 6
9 4

SAMPLE OUTPUT:

9

FJ can create four balanced towers with cows of weights 5 and 9, and one balanced tower with a cow of weight 7. Alternatively, he can create four balanced towers with cows of weights 5 and 9, and one balanced tower with a cow of weight 5.

SCORING: In inputs 3-5, $M \leq 5000$ and the total number of cows does not exceed $5000$.In inputs 6-11, the total number of cows does not exceed $2\cdot 10^5$.Inputs 12-17 have no additional constraints.

Problem credits: Eric Hsu