Chain

Consider an isothetic rectangle (where "isothetic" means "with sides parallel to coordinate axes"), left bottom corner has coordinates $(0,0)$, right top one — (x_{\max}, y_{\max}). There are $N$ points P_1 (x_1, y_1), P_2 (x_2, y_2), …, P_N (x_N, y_N) inside the rectangle (0 < x_i < x_{\max}, 0 < y_i < y_{\max}). Consider polygonal chains SP_{i1}P_{i2}...P_{ik}F satisfying the following constraints:

1. All internal vertices P_{ij} are some of the given points; quantity of selected points can be any 0 \leq k \leq N;

2. Start vertex $S$ (x_S, y_S) is located on the left side of the rectangle or outside it to the left (x_S \leq 0, 0 \leq y_S \leq y_{\max});

3. End vertex $F$ (x_F, y_F) is located on the right side of the rectangle or outside it to the right (x_F \geq x_{\max}, 0 \leq y_F \leq y_{\max}).

Let's call such chain constant-length if and only if Euclidean lengths of all segments are equal (|SP_{i1}| = |P_{i1}P_{i2}| = ... = |P_{ik}F|). It's easy to see that there is at least one constant-length polygonal chain for any rectangle and any set of given points: this is segment $SF$ with coordinates S(0, 0), F(x_{\max}, 0) which satisfies all the constraints.

Your task is to write a program to find a constant-length chain with minimal length of segment. It is easy to see that square (second degree) of the minimal length is always integer, so program should output square of the length, as integer number.

Sortie

Your program should write exactly one integer: square of the minimal possible segment length among constant-length polygonal chains.