Circumferences

AtCoder

IDabc259_d

Time2000ms

Memory256MB

Difficulty—

You are given $N$ circles on the $xy$\-coordinate plane. For each $i = 1, 2, \ldots, N$, the $i$\-th circle is centered at $(x_i, y_i)$ and has a radius of $r_i$. Determine whether it is possible to get from $(s_x, s_y)$ to $(t_x, t_y)$ by only passing through points that lie on the circumference of at least one of the $N$ circles. ## Constraints * $1 \leq N \leq 3000$ * $-10^9 \leq x_i, y_i \leq 10^9$ * $1 \leq r_i \leq 10^9$ * $(s_x, s_y)$ lies on the circumference of at least one of the $N$ circles. * $(t_x, t_y)$ lies on the circumference of at least one of the $N$ circles. * All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ $s_x$ $s_y$ $t_x$ $t_y$ $x_1$ $y_1$ $r_1$ $x_2$ $y_2$ $r_2$ $\vdots$ $x_N$ $y_N$ $r_N$ [samples]

Samples

Input #1

Output #1

Yes

![image](https://img.atcoder.jp/abc259/7b850385b9d67dc150435ffc7818bd94.png)
Here is one way to get from $(0, -2)$ to $(3, 3)$.

*   From $(0, -2)$, pass through the circumference of the $1$\-st circle counterclockwise to reach $(1, -\sqrt{3})$.
*   From $(1, -\sqrt{3})$, pass through the circumference of the $2$\-nd circle clockwise to reach $(2, 2)$.
*   From $(2, 2)$, pass through the circumference of the $3$\-rd circle counterclockwise to reach $(3, 3)$.

Thus, `Yes` should be printed.

Input #2

Output #2

No

![image](https://img.atcoder.jp/abc259/924efa40ff28e5d7125841da2710d012.png)
It is impossible to get from $(0, 1)$ to $(0, 3)$ by only passing through points on the circumference of at least one of the circles, so `No` should be printed.

API Response (JSON)

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      "content": "You are given $N$ circles on the $xy$\\-coordinate plane. For each $i = 1, 2, \\ldots, N$, the $i$\\-th circle is centered at $(x_i, y_i)$ and has a radius of $r_i$. Determine whether it is possible to g",
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      "statement_type": "Markdown",
      "content": "You are given $N$ circles on the $xy$\\-coordinate plane. For each $i = 1, 2, \\ldots, N$, the $i$\\-th circle is centered at $(x_i, y_i)$ and has a radius of $r_i$.\nDetermine whether it is possible to g...",
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