Lasers and Mirrors

The cows are putting on another laser light show.

The laser they got for this show is so large that they cannot move it from the spot where it was delivered. They want to send its beam to the barn on the other side of the farm. On the map of the farm the laser and the barn are points in the plane. The cows aim the laser so that the beam leaves horizontally or vertically, parallel to the x axis or the y axis. They then bounce the beam off a few mirrors to bring it to the barn.

The farm has $N$ fence posts ($1 \le N \le 100{,}000$) standing at distinct points. No post stands where the laser or the barn is. The cows can mount a mirror on any posts they like. A post with no mirror lets the beam pass straight over it without changing direction. A post with a mirror holds it diagonally, as / or \, so it turns a horizontal beam into a vertical one and a vertical beam into a horizontal one. The cows choose the orientation of each mirror.

Compute the smallest number of mirrors that brings the beam from the laser to the barn.

The first line contains five space separated integers $N$, $x_L$, $y_L$, $x_B$, $y_B$, where $(x_L, y_L)$ is the location of the laser and $(x_B, y_B)$ is the location of the barn.

Each of the next $N$ lines contains the coordinates $x$ and $y$ of one fence post. Every coordinate is an integer between $0$ and $1{,}000{,}000{,}000$, inclusive. The $N$ posts are at distinct points, and none of them is at the laser or at the barn.

Print the smallest number of mirrors needed to send the beam to the barn. If no arrangement of mirrors sends the beam to the barn, print -1.