#215815. [JAG 2025 Summer Camp #1] Pole

On the sphere centered at with radius , there are circles. The -th circle is defined as the set of intersection points between the sphere and the following plane:

• The plane passes through the point .
• The plane is orthogonal to .

It is guaranteed that .

It is also guaranteed that no two circles share any common point.

:::align{center}

Figure E-1: The circle defined by :::

We define the distance between two points on the sphere as follows:

For a path on the sphere connecting these two points, count how many of the circles the path intersects. The distance is the minimum possible value of this count over all such paths.

You may choose one point on the sphere and designate it as the Pole. Find the minimum possible value of

The input is given in the following format:

• ()
• No two circles share any common point.
• All input values are integers.

Output the answer in a single line.

样例输入 1

5 100
14 -11 -2
-54 46 8
54 -57 -12
-34 39 7
64 -74 -4

样例输出 1

3

:::align{center}

Figure E-2: Illustration of Sample Input :::