Dog & Gopher

A large field contains a dog and a gopher. The dog wants to eat the gopher, while the gopher wants to reach safety through one of several gopher holes dug in the field.

Neither animal is a mathematician, but neither is foolish either. The gopher picks one hole and runs toward it in a straight line at a fixed speed $v$. The dog reads the gopher's body language, instantly figures out which hole the gopher chose, and runs in a straight line toward that same hole at twice the gopher's speed, $2v$. If the dog reaches the hole first, it eats the gopher; otherwise the gopher escapes. If both reach the hole at the same instant, the gopher is considered to escape.

You are helping the gopher: decide whether it can escape through some hole.

The first line contains four floating-point numbers: the $(x, y)$ coordinates of the gopher followed by the $(x, y)$ coordinates of the dog. Each of the following lines contains two floating-point numbers: the $(x, y)$ coordinates of one gopher hole. All distances are in metres, given to the nearest millimetre. There are at most $1000$ gopher holes, and every coordinate is between $-10000$ and $10000$.

Print a single line. Consider the gopher holes in the order they appear in the input. The gopher can escape through a hole if it reaches that hole no later than the dog, that is, if twice the gopher-to-hole distance is at most the dog-to-hole distance.

If at least one such hole exists, output the first one in input order in the form:

The gopher can escape through the hole at (x,y).

where x and y are that hole's coordinates printed to the nearest millimetre (three decimal places). If no hole lets the gopher escape, output:

The gopher cannot escape.