Bingo

Bingo is a game played on a square grid. Each player gets an $n \times n$ grid and writes a distinct number in each cell. The host then draws a random number, and each player looks for that number on their grid and fills the corresponding cell if the number is present. This repeats until someone gets $n$ filled cells on a single line, which we call a bingo line.

There are $2n + 2$ possible bingo lines: $n$ horizontal lines, $n$ vertical lines, and $2$ diagonal lines.

---  ...  ...  |..  .|.  ..|  \..  ../
...  ---  ...  |..  .|.  ..|  .\.  ./.
...  ...  ---  |..  .|.  ..|  ..\  /..

For example, the following grid has four bingo lines: two horizontal lines, one vertical line, and one diagonal line.

#..#.
#####
..###
#####
..###

When exactly is a bingo line formed? This is completely random. If you are lucky you can complete a line quite early, and on the other hand you can fill most of the grid without making any bingo line. In this problem we look at the unlucky case of filling $k$ cells without making any bingo line.

Given two integers $n$ and $k$, determine whether it is possible to fill exactly $k$ cells of an $n \times n$ grid without making any bingo line. If it is possible, show one way to do it.

The first and only line of input contains two integers, $n$ and $k$.

On the first line, output YES if it is possible to fill exactly $k$ cells of an $n \times n$ grid without making any bingo line. Otherwise, output NO.

If the answer is YES, output each row of the grid starting from the next line. Each row is a string of $n$ characters. The $i$-th character is # (ASCII 35) if the $i$-th cell of the row is filled, and . (ASCII 46) if it is not. Exactly $k$ cells must be filled, and there must be no bingo line.

If there are multiple ways to fill the grid, output any one of them.

• $1 \leq n \leq 100$
• $0 \leq k \leq n^2$

The second example is valid only for subtasks 2 and 3.