Hill Walk

There are $N$ hills ($1 \le N \le 100{,}000$). Each hill is a line segment from $(x_1, y_1)$ to $(x_2, y_2)$ with $x_1 < x_2$ and $y_1 < y_2$. No two segments intersect or even touch, not even at their endpoints, and the first hill satisfies $(x_1, y_1) = (0, 0)$.

Bessie the cow starts at $(0, 0)$ on the first hill. Whenever Bessie is on a hill, she climbs up until she reaches its upper end, then jumps off the edge. If she lands on another hill she keeps walking along that hill; otherwise she falls forever and lands safely on a cushion of pillows at $y = -\infty$.

Treat each hill from $(x_1, y_1)$ to $(x_2, y_2)$ as containing the point $(x_1, y_1)$ but not the point $(x_2, y_2)$: if Bessie falls straight down at $x = x_1$ she lands on the hill, but if she falls at $x = x_2$ she does not.

Count the total number of hills Bessie touches at some point during her walk.

• Line $1$: the number of hills, $N$.
• Lines $2 \ldots N+1$: line $i+1$ contains four integers $x_1\ y_1\ x_2\ y_2$ describing hill $i$. Every integer is in the range $0 \ldots 1{,}000{,}000{,}000$.

• Line $1$: the number of hills Bessie touches during her journey.

In the example, there are four hills. The first hill runs from $(0, 0)$ to $(5, 6)$. Starting on it, Bessie walks along hills #1, #4, and finally #3, touching three hills in total.