Dungeon Escape

Brave Sir Robin has been imprisoned in a dungeon by an evil king. The dungeon has infinitely many cube-shaped rooms separated by thick stone walls. Consecutive rooms are connected by passages, and when viewed from above the rooms form the following square spiral.

After a large earthquake, some walls collapsed and additional passages opened between adjacent rooms that had previously been separated by a wall.

Sir Robin starts in room 1. The dungeon exit is in room N. Because a dragon guards the dungeon, Sir Robin wants to leave using as few passages as possible.

Given N and the list of newly opened passages, determine the minimum number of passages Sir Robin must traverse to reach the exit.

The first line contains an integer N (1 <= N <= 10^15), the room containing the exit.

The second line contains an integer K (1 <= K <= 100000), the number of newly opened passages.

Each of the next K lines contains an integer B (4 <= B <= 10^15). This means that a new passage connects two adjacent rooms A and B, where A < B. The value A is not given, but it is uniquely determined by B. For instance, if B is 20, then A is 7. Some rooms, such as 2, 3, 5, 7, 10, and 13, can never appear as B.

Print one integer: the minimum number of passages Sir Robin must traverse to reach the exit.

For the first visible test, the dungeon after the earthquake looks like this.

One shortest route is 1-4-15-14-13-30-31, which uses 6 passages.