Problem K
Fire Hydrant

Firefighter Robert has been working as a firefighter for over 20 years and loves his job. After a long walk in the metropolis M (formerly known as Stockholm), Robert received a call on his phone from his hydrophobic chief, ordering him to come to the fire station right away. Robert has just arrived at the most south-west four-way intersection in metropolis M and decides to head straight to the fire station, but apparently, several fire hydrants in metropolis M have started leaking water and flooding the streets! Oh no! This is particularly serious because Robert’s hydrophobic chief at the fire station is extremely strict about not having water near him, so showing up soaking wet to work after encountering several of the leaking fire hydrants will get Robert fired. Robert wants to work as a firefighter for at least 20 more years, so help Robert minimize how wet he gets!

Metropolis M consists of a grid with dimensions $W \times H$, where each four-way intersection is a square in the grid, and is reachable from the four adjacent intersections in each cardinal direction: north, east, south, and west. Robert has just arrived at the four-way intersection furthest southwest in the grid, at position $(1,1)$. Robert wants to reach the fire station furthest northeast in the grid, at position $(W,H)$. It takes exactly 1 minute to run from one four-way intersection to another adjacent intersection. It is guaranteed that the fire station is not located at $(1,1)$.

Furthermore, there are $N$ four-way intersections with a leaking fire hydrant, which can be located in any square on the grid, including where Robert starts from $(1,1)$ and at the fire station $(W,H)$. Robert arrives at the square $(1,1)$ at minute 0, and each leaking fire hydrant has leaked one unit of water to the intersection the fire hydrant is located at. For each minute $t$ that passes, each leaking fire hydrant will leak an additional unit of water to all intersections within a certain distance from it. Specifically, all squares $(x,y)$ that satisfy the following inequality at minute $t$ will increase by one unit of water from the fire hydrant located at $(x_ b,y_ b)$:

Let’s call the number of units of water at a particular square $v_ t$ at minute $t$. When Robert arrives at a four-way intersection at minute $t$, Robert’s clothes will get wet from all the water present at that intersection from leaking fire hydrants, which is $v_ t$ units of water.

What is the minimum total number of water units that Robert needs to go through on his way to the fire station?

Figure 1: Illustration of sample 1, where the green square is Roberts position and the red squares are the positions of the fire hydrants. The smallest total water units Bob can have when arriving at the station is $0+0+2+2+4+6+8+8=30$.

Input

The first line of the input contains three integers $W,H,N$ ($1 \le W,H \le 1000$, $1 \le N \le min(20000,W \cdot H)$), where $W$ and $H$ are the width and height of the grid and $N$ is the number of leaking fire hydrants.

The following $N$ rows each contain two integers $x_ i$ and $y_ i$ ($1 \le x_ i \le W$, $1 \le y_ i \le H$), where $(x_ i,y_ i)$ is the position of the $i$:th leaking fire hydrant. It is guaranteed that no two fire hydrants occupy the same cell.

Output

Print an integer, the minimum number of water units Robert has to pass through on his way to the fire station.

Points

Your solution will be tested on several test case groups. To get the points for a group, it must pass all the test cases in the group.

5 4 2
4 1
2 3

30

2 2 1
2 2

4