Lamps

The architect Ziemobit designed a glass corridor connecting two government offices. Several lamps hang from the corridor's ceiling, and each lamp shines light straight down in a cone with an apex angle of $90^\circ$ (a triangle seen from the side, spreading $45^\circ$ to each side of the point directly below the lamp).

Seen from the side, near the ceiling between any two neighboring lamps there is a triangular dark region that neither lamp's light reaches.

The corridor is tall enough that the original lamps already light the entire floor, and there are already lamps at both ends of the corridor (position $0$ and position $D$). So the only dark regions are the triangles between neighboring lamps.

There is budget left to add up to $K$ more lamps at any positions on the ceiling. Adding lamps shrinks the dark region. After adding at most $K$ lamps, make the total area of the dark region (seen from the side) as small as possible, and report that minimum.

The first line contains three integers $N$, $K$, and $D$ ($2 \le N \le 100\,000$, $0 \le K \le 100\,000$, $1 \le D \le 10^9$): the number of lamps already hanging, the number of lamps you may add, and the length of the corridor.

The second line contains $N$ increasing integers giving the lamp positions. The first is $0$ and the last is $D$.

Print the minimum total area of the dark region after adding at most $K$ lamps, as a reduced fraction $p/q$ where $p$ and $q$ are integers with $q \ge 1$ and $\gcd(p, q) = 1$.