Eric is building his own Cereal City, and he needs your help planning it out!

Cereal City can be presented by a unit grid with $$$n+2$$$ vertical lines (numbered $$$0$$$ to $$$n+1$$$ from left to right), and $$$n+2$$$ horizontal lines (numbered $$$0$$$ to $$$n+1$$$ from top to bottom). Let $$$(x, y)$$$ represent the intersection between the $$$x$$$-th horizontal line and $$$y$$$-th vertical line.

There are $$$m$$$ buildings in this city, with the $$$i$$$-th building is located at ($$$x_i, y_i$$$) such that $$$1 \leq x_i, y_i \leq n$$$.

Example of a Cereal City where $$$n=2, m=2,$$$ and there is a building at $$$(1, 2)$$$ and $$$(2, 1)$$$.

You want to build some roads along the grid lines such that you can reach any city from any other city via some series of roads. However, Eric wants you to build the roads in a peculiar method:

You want to start at one of the 4 borders of the grid, and build inwards perpendicular to the border along a grid line until you reach an intersecting road or the opposite border. You may not build past this point. The amount of road you use is equivalent to the number of intersections that lies on the road minus $$$1$$$. Furthermore, no roads are to be built along the border of the grid (the $$$0$$$-th or $$$n+1$$$-th horizontal or vertical line).

Note that the order that you build the roads matters.

Some examples of valid / invalid road formations.

You have all the time you need to build the roads, but the deadline for ordering supplies is coming fast! Please tell Eric the minimum amount of road you need to connect all cities, so that he can order the supplies!