Finding Treasure

Morgan is designing a treasury for a treasure hunt event. The treasury can be represented as one dimensional grid with $N + 2$ cells, numbered from $0$ to $N + 1$. Each cell can be empty or contain a treasure. At first, cell $0$ and cell $N + 1$ contain a treasure, while the other cells are currently empty.

While experimenting with the treasury design, Morgan does $Q$ updates, numbered from $1$ to $Q$, to his treasury. In update $i$, he wants to change cell $A\_i$. If cell $A\_i$ is empty, he will put a treasure on cell $A\_i$. If cell $A\_i$ contains a treasure, he will remove the treasure from cell $A\_i$ and the cell becomes empty. It is guaranteed that $1 ≤ A\_i ≤ N$ for all $1 ≤ i ≤ Q$, which implies cell $0$ and cell $N + 1$ always contain a treasure.

After each change, Morgan wonders how difficult his treasure hunt is. He defines the difficulty level of standing on cell $x$ as the multiplication of two values:

• the distance from $x$ to the closest treasure on cell $y ≤ x$, and
• the distance from $x$ to the closest treasure on cell $y ≥ x$.

The distance between two cells can be calculated as the absolute difference between the cell numbers. Then, the total difficulty level of his treasure hunt is defined as the sum of difficulty level of standing on cell $x$ for all $0 ≤ x ≤ N + 1$.

Help Morgan to determine the total difficulty level of his treasure hunt after each update.

Input begins with two integers $N$ $Q$ ($1 ≤ N ≤ 100\\, 000$; $1 ≤ Q ≤ 100\\, 000$) representing the size of the treasury room and the number of updates, respectively. Each of the next $Q$ lines contains an integer $A\_i$ ($1 ≤ A\_i ≤ N$) representing the cell that Morgan wants to change in update $i$.

After each update that Morgan makes, output an integer in a single line representing the total difficulty level of his treasure hunts at that time.