Mowing the Field

Farmer John runs his farm well in almost every way, except that he is always late with the mowing. He moves the mower only once a day. On day 1 he starts at $(x_1, y_1)$, and on day $d$ he mows along a straight segment from the previous day's position to $(x_d, y_d)$. On the 2D map of the farm every move is horizontal or vertical, so $x_d = x_{d-1}$ or $y_d = y_{d-1}$. John alternates horizontal and vertical moves on successive days.

He works so slowly that grass he already cut grows back before he is done. Grass cut on day $d$ reappears on day $d + T$, so whenever his path meets a segment he cut at least $T$ days earlier, he cuts the same spot a second time. John wants to know how bad his routine is, so he wants to count how often that happens.

Count the crossing points where John re-cuts grass that had grown back. Only perpendicular meetings count, meaning a point shared by a horizontal segment and a vertical segment that is an endpoint of neither segment.

The first line contains $N$ ($2 \le N \le 100\,000$) and $T$ ($1 \le T \le N$, $T$ even).

Each of the next $N$ lines gives the position of the mower on days $1$ through $N$. Line $i$ contains the integers $x_i$ and $y_i$, both between $0$ and $10^9$.

Two consecutive positions may be equal. The move on such a day is a single point and takes part in no crossing.

Print the number of crossing points described above.

In the first example the path of day 7 meets the segment cut on day 2. The two days are $T = 4$ or more apart, so this crossing counts. The other two crossings are only 3 days apart and do not count.