## USACO 2024 US Open Contest, Platinum

## Problem 3. Activating Robots

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You and a single robot are initially at point $0$ on a circle with perimeter $L$ ($1 \le L \le 10^9$). You can move either counterclockwise or clockwise along the circle at $1$ unit per second. All movement in this problem is continuous.

Your goal is to place exactly $R-1$ robots such that at the end, every two consecutive robots are spaced $L/R$ away from each other ($2\le R\le 20$, $R$ divides $L$). There are $N$ ($1\le N\le 10^5$) activation points, the $i$th of which is located $a_i$ distance counterclockwise from $0$ ($0\le a_i<L$). If you are currently at an activation point, you can instantaneously place a robot at that point. All robots (including the original) move counterclockwise at a rate of $1$ unit per $K$ seconds ($1\leq K\leq 10^6$).

Compute the minimum time required to achieve the goal.

#### INPUT FORMAT (input arrives from the terminal / stdin):

The first line contains $L$, $R$, $N$, and $K$.The next line contains $N$ space-separated integers $a_1,a_2,\dots,a_N$.

#### OUTPUT FORMAT (print output to the terminal / stdout):

The minimum time required to achieve the goal.#### SAMPLE INPUT:

10 2 1 2 6

#### SAMPLE OUTPUT:

22We can reach the activation point at $6$ in $4$ seconds by going clockwise. At this time, the initial robot will be located at $2$. Wait an additional $18$ seconds until the initial robot is located at $1$. Now we can place a robot to immediately win.

#### SAMPLE INPUT:

10 2 1 2 7

#### SAMPLE OUTPUT:

4

We can reach the activation point at $7$ in $3$ seconds by going clockwise. At this time, the initial robot will be located at $1.5$. Wait an additional second until the initial robot is located at $2$. Now we can place a robot to immediately win.

#### SAMPLE INPUT:

32 4 5 2 0 23 12 5 11

#### SAMPLE OUTPUT:

48

#### SAMPLE INPUT:

24 3 1 2 16

#### SAMPLE OUTPUT:

48

#### SCORING:

- Inputs 5-6: $R=2$
- Inputs 7-12: $R\le 10, N\le 80$
- Inputs 13-20: $R\le 16$
- Inputs 21-24: No additional constraints.

Problem credits: Benjamin Qi