LaLa and Monster Hunting (Part 1)

A dreadful monster has been witnessed in a forest near the city of $\color{red}{\text{m}} \color{brown}{\text{a}} \color{orange}{\text{g}} \color{blue}{\text{i}} \color{magenta} {\text{c}}$ Sharia, and a group of valorous adventurers will hunt it down in few days before it hurts anyone. However, $\color{blue}{\text{LaLa}}$ knows that the real reason those adventurers are willing to take the risk is to obtain the rare $\color{red}{\text{m}} \color{brown}{\text{a}} \color{orange}{\text{g}} \color{blue}{\text{i}} \color{magenta} {\text{c}}$ stone that the monster is known to produce in its intestines. $\color{blue}{\text{LaLa}}$ would like to obtain the $\color{red}{\text{m}} \color{brown}{\text{a}} \color{orange}{\text{g}} \color{blue}{\text{i}} \color{magenta} {\text{c}}$ stone before they do, as it is known to be quite beautiful.

$\color{blue}{\text{LaLa}}$ will first locate the monster with her $\color{red}{\text{m}} \color{brown}{\text{a}} \color{orange}{\text{g}} \color{blue}{\text{i}} \color{magenta} {\text{c}}$. $\color{blue}{\text{LaLa}}$ has placed a bunch of $\color{red}{\text{m}} \color{brown}{\text{a}} \color{orange}{\text{g}} \color{blue}{\text{i}} \color{magenta} {\text{c}}$ tools within the forest, each of which has some power associated with it.

Consider the circles centered at each $\color{red}{\text{m}} \color{brown}{\text{a}} \color{orange}{\text{g}} \color{blue}{\text{i}} \color{magenta} {\text{c}}$ tool with radius equal to its power. $\color{blue}{\text{LaLa}}$'s $\color{red}{\text{m}} \color{brown}{\text{a}} \color{orange}{\text{g}} \color{blue}{\text{i}} \color{magenta} {\text{c}}$ will successfully locate the monster if and only if the convex hull of the circles contains the location of the monster.

Write a program that determines whether $\color{blue}{\text{LaLa}}$ will be able to locate the monster.

The input is given in the following format:

$N$

$x\_0$ $y\_0$ $r\_0$

$x\_1$ $y\_1$ $r\_1$

$\vdots$

$x\_{N-1}$ $y\_{N-1}$ $r\_{N-1}$

where $N$ is the number of $\color{red}{\text{m}} \color{brown}{\text{a}} \color{orange}{\text{g}} \color{blue}{\text{i}} \color{magenta} {\text{c}}$ tools placed in the forest, the $i$-th of which is located at $(x\_i, y\_i)$ and has power $r\_i$. Here, assume that the forest is a two-dimensional plane where the monster is located at $(0, 0)$.

The input satisfies the following constraints:

• All the numbers in the input are integers.
• $1 \le N \le 1\\,000\\,000$
• $-1\\,000\\,000 \le x\_i, y\_i \le 1\\,000\\,000$ for all integers $0 \le i < N$
• $0 \le r\_i \le 1\\,000\\,000$ for all integers $0 \le i < N$
• The distance between point $(0, 0)$ and the boundary of the convex hull of $N$ circles, $i$-th of which is centered at $(x\_i, y\_i)$ and has radius $r\_i$, is at least $1$.

If $\color{blue}{\text{LaLa}}$'s $\color{red}{\text{m}} \color{brown}{\text{a}} \color{orange}{\text{g}} \color{blue}{\text{i}} \color{magenta} {\text{c}}$ will successfully locate the monster, print a single string "Yes". Otherwise, print a single string "No". You may print each character in either case (lower or upper).

The following illustrates the configuration of the $\color{red}{\text{m}} \color{brown}{\text{a}} \color{orange}{\text{g}} \color{blue}{\text{i}} \color{magenta} {\text{c}}$ tools for the sample tests. The red curve denotes the boundary of the convex hull.

First Sample	Second Sample	Third Sample