## USACO 2024 February Contest, Gold

## Problem 1. Bessla Motors

Contest has ended.

**Log in to allow submissions in analysis mode**

****Note: The time limit for this problem is 3s, 1.5x the default. The memory
limit for this problem is 512MB, twice the default.****

Farmer John would like to promote his line of Bessla electric tractors by showcasing Bessla's network of charging stations. He has identified $N$ ($2\le N\le 5\cdot 10^4$) points of interest labeled $1\dots N$, of which the first $C$ ($1\le C < N$) are charging stations and the remainder are travel destinations. These points of interest are interconnected by $M$ ($1\le M\le 10^5$) bidirectional roads, the $i$-th of which connects distinct points $u_i$ and $v_i$ ($1\le u_i, v_i\le N$) and has length $\ell_i$ miles ($1\le\ell_i\le 10^9$).

A Bessla can travel up to $2R$ miles ($1\le R\le 10^9$) on a single charge,
allowing it to reach any destination within $R$ miles of a charging station. A
destination is deemed *well-connected* if it is reachable from at least $K$
($1\le K\le 10$) distinct charging stations. Your task is to assist Farmer John
in identifying the set of well-connected travel destinations.

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

The first line contains five space-separated integers $N$, $M$, $C$, $R$, and $K$. Each of the following $M$ lines contains three space-separated integers $u_i$, $v_i$, and $\ell_i$ such that $u_i\neq v_i$.The charging stations are labeled $1, 2, \ldots, C$. The remaining points of interest are all travel destinations.

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

First, output the number of well-connected travel destinations on a single line. Then, list all well-connected travel destinations in ascending order, each on a separate line.#### SAMPLE INPUT:

3 3 1 4 1 1 2 3 1 3 5 2 3 2

#### SAMPLE OUTPUT:

1 2

We have one charging station at $1$. From this charging station, we can reach point $2$ (since it is distance $3$ away from $1$), but not point $3$ (since it is distance $5$ away from $1$). Thus, only point $2$ is well-connected.

#### SAMPLE INPUT:

4 3 2 101 2 1 2 1 2 3 100 1 4 10

#### SAMPLE OUTPUT:

2 3 4

We have charging stations at $1$ and $2$, and both points $3$ and $4$ are within distance $101$ of both $1$ and $2$. Thus, both points $3$ and $4$ are well-connected.

#### SAMPLE INPUT:

4 3 2 100 2 1 2 1 2 3 100 1 4 10

#### SAMPLE OUTPUT:

1 4

#### SCORING:

- Inputs 4 and 5: $K = 2$ and $N \le 500$ and $M\le 1000$.
- Inputs 6 and 7: $K = 2$.
- Inputs 8-15: No additional constraints.

Problem credits: Alexander Wei