Firing the Phaser

You are the captain of a spaceship and have snuck up on the most fierce enemy you have ever faced. You plan to fire a large phaser cannon at the enemy flagship before it notices you. There is no room for error, so the shot must be perfect.

You have started charging the phaser and pulled the room layout of the flagship from the archives. You are directly above the enemy, and the layout is a two-dimensional map of rooms. Each room is a rectangle with sides parallel to the x and y axes, and no two rooms intersect, not even at a single point.

The phaser beam is set by a point $(x, y)$ and an angle $\vartheta$. The beam starts at $(x, y)$ and travels a distance $\ell$ in the direction $\vartheta$, damaging every room it touches. Your goal is to hit as many rooms as possible with one shot.

The charge is almost complete, and the only thing left is choosing the best setting. That turns out to be harder than expected. You have ten seconds until the charge finishes, so you write a program to solve it.

The first line contains two integers $r$ and $\ell$ ($1 \le r \le 15$, $1 \le \ell \le 1000$), where $r$ is the number of rooms in the flagship and $\ell$ is the length of a phaser shot.

The next $r$ lines each contain four integers $x_1$, $y_1$, $x_2$, $y_2$ ($0 \le x_1 < x_2 \le 1000$, $0 \le y_1 < y_2 \le 1000$). Each line describes a room whose lower left corner is $(x_1, y_1)$ and whose upper right corner is $(x_2, y_2)$.

Output one line with the maximum number of rooms that one phaser beam can hit. A room counts as hit if the beam touches it.

You may assume the answer is numerically stable: if every room is expanded by $10^{-6}$ in all four directions, the answer does not change.