A mapping company is testing a new signal simulation system. They have placed $$$N$$$ signal towers at different positions on a flat map. Each tower is located at coordinates $$$(x_i, y_i)$$$, where $$$1 \le x_i, y_i \le 2 \cdot 10^5$$$, all $$$y_i$$$ values are distinct and all $$$x_i$$$ values are distinct as well, additionally, there are no three colinear points.

To analyze signal interference, the company draws a virtual line between every pair of towers. These lines are extended downward, and the engineers are interested in how these lines affect positions on a distant horizontal scan line at $$$y = -10^{18}$$$.

They define a function $$$f(x)$$$ for real numbers $$$x$$$, which represents interference at horizontal position $$$x$$$ on the scan line. Initially, $$$f(x) = 0$$$ for all $$$x$$$.

Then, for every pair of towers $$$i \lt j$$$, the following process is applied:

• Consider the directed line from point $$$i$$$ to point $$$j$$$.
• For a fixed real number $$$x$$$, consider the point $$$(x, -10^{18})$$$.
  • If this point is on or to the left of the line (remember that the line is directed from point $$$i$$$ to point $$$j$$$), then update $$$f(x) := f(x) \oplus a$$$.
  • If it is to the right of the line, then update $$$f(x) := f(x) \oplus b$$$.

Here, $$$\oplus$$$ denotes the bitwise XOR operation.

After processing all pairs, the function $$$f(x)$$$ becomes piecewise constant: the real line is divided into a number of intervals where $$$f(x)$$$ has the same value. We define a Trujo as any such interval where $$$f(x) = 0$$$.

Your task is to count how many Trujos appear on the real number line.