Area of Effect

Liam plays a tower defense game. He destroys his opponent's minions while defending his own villages.

Liam's favorite attack is an area of effect attack. The attack range is a perfect circle. Liam picks a center and a radius, and every minion inside that circle or on it is destroyed. A minion is a point with no size.

The attack must not damage Liam's villages. A village is also a perfect circle. The attack circle may touch the wall of a village, but it must not reach inside the village. For a village with center $(v_x, v_y)$ and radius $v_r$, and an attack with center $(c_x, c_y)$ and radius $\rho$, the distance between the two centers must be at least $\rho + v_r$.

The attack radius has an upper limit $r$. Liam can attack with a radius smaller than the limit, but never with a larger one.

Find the largest number of minions that one attack destroys without touching any village.

The first line contains three integers $n$, $m$, $r$ separated by spaces.

• $n$ ($1 \le n \le 10$) is the number of villages.
• $m$ ($1 \le m \le 2000$) is the number of opposing minions.
• $r$ ($1 \le r \le 20000$) is the upper limit of the attack radius.

Each of the next $n$ lines contains three integers $v_x$, $v_y$, $v_r$ describing one village. $(v_x, v_y)$ is its center ($-20000 \le v_x, v_y \le 20000$) and $v_r$ is its radius ($1 \le v_r \le 20000$). No two villages intersect or overlap.

Each of the next $m$ lines contains two integers $m_x$, $m_y$, the position of one minion ($-20000 \le m_x, m_y \le 20000$). No two minions share a position, and no minion is inside a village. A minion can be on the wall of a village.

Print one integer, the largest number of minions destroyed by a single attack.