Nice Lines

Roxette the pirate princess has arrived to the secret island in the Remeian archipelago. There, a famous treasure, the golden nice lines is rumoured to be buried.

The secret island is a square, $2 × 10^{12}$ by $2 × 10^{12}$ meters long and tall. Any point on the island is described using Cartesian coordinates, with $(0, 0)$ being at the center, and the two axes being parallel to its sides.

There are $N$ golden nice lines buried on the island. The $i$th one for $0 ≤ i < N$ occupies the set of all real-valued points $(x, y)$ described by the linear equation $y = a\_ix + b\_i$.

Roxette can use a special device, called a lineometer. Given any point $p$ on the island, the lineometer will compute the sum of the distances1 from point $p$ to each of the $N$ golden nice lines.

Unfortunately, the lineometer has a limited number of uses. Can you help Roxette find the treasure with a small enough number of lineometer uses?

1The Euclidean distance between a point and a line is the length of the shortest line segment that touches both the line and the point.

• $1 ≤ N ≤ 100$
• $−10\\,000 ≤ a\_i , b\_i ≤ 10\\,000$
• No two lines are parallel.