Ritual Circle

Before the Fellowship set out from Rivendell, Bilbo gave Frodo the Elvish blade he called Sting. The sword is special: its blade glows blue whenever Orcs draw near.

Frodo wants to draw a single circle that gathers all of his companions safely inside while keeping every Orc outside. Help him find the smallest such circle.

The input contains several test cases, one after another, until the end of file. Each test case is given on two lines:

• A line that begins with Companions: followed by the positions of Frodo's companions.
• A line that begins with Orcs: followed by the positions of the Orcs.

Each position is written as (x,y) with integer coordinates, and a line lists zero or more points separated by spaces. The word none denotes an empty set. All coordinates are integers between $0$ and $100$ inclusive, and every point within a single test case is distinct. Across both groups a test case has at most $300$ points in total, and at most $10$ test cases contain more than $200$ points.

For each test case, consider the smallest circle that contains every companion position (a companion may lie on the boundary) while every Orc stays strictly outside it.

If no such circle exists, print The Orcs are close.

Otherwise print the square of that smallest circle's radius as an exact fraction in lowest terms, written p/q with q >= 1 (print 0/1 when the radius is $0$). The squared radius is always rational because the optimal circle is pinned by integer-coordinate points.

The smallest radius can be an infimum that is not actually attained: sometimes the tightest circle would place an Orc exactly on its boundary, yet a circle whose center is nudged by an infinitesimal amount and whose radius is infinitesimally larger keeps that Orc strictly outside. In that case still report the squared radius of the limiting circle. When a test case has a single companion, a circle of radius $0$ already works, so the answer is 0/1.