The city of Fishtopia can be imagined as a grid of $4$ rows and an **odd** number of columns. It has two main villages; the first is located at the top-left cell $(1,1)$, people who stay there love fishing at the Tuna pond at the bottom-right cell $(4, n)$. The second village is located at $(4, 1)$ and its people love the Salmon pond at $(1, n)$.
The mayor of Fishtopia wants to place $k$ hotels in the city, each one occupying one cell. To allow people to enter the city from anywhere, hotels should not be placed on the border cells.
A person can move from one cell to another if those cells are not occupied by hotels and share a side.
Can you help the mayor place the hotels in a way such that there are equal number of shortest paths from each village to its preferred pond?
## Input
The first line of input contain two integers, $n$ and $k$ ($3 \leq n \leq 99$, $0 \leq k \leq 2\times(n-2)$), $n$ is **odd**, the width of the city, and the number of hotels to be placed, respectively.
## Output
Print "_YES_", if it is possible to place all the hotels in a way that satisfies the problem statement, otherwise print "_NO_".
If it is possible, print an extra $4$ lines that describe the city, each line should have $n$ characters, each of which is "_#_" if that cell has a hotel on it, or "_._" if not.
[samples]
鱼拓城可以被想象为一个 4 行、列数为 *奇数* 的网格。该城有两个主要村庄：第一个村庄位于左上角单元格 $(1, 1)$，那里的居民喜欢到右下角单元格 $(4, n)$ 的金枪鱼池钓鱼；第二个村庄位于 $(4, 1)$，其居民喜欢位于 $(1, n)$ 的鲑鱼池。
鱼拓城的市长希望在城市中放置 $k$ 家酒店，每家酒店占据一个单元格。为了使人们可以从任何地方进入城市，酒店不能放置在边界单元格上。
一个人可以从一个单元格移动到另一个单元格，当且仅当这两个单元格均未被酒店占据且共享一条边。
你能否帮助市长安排酒店的位置，使得从每个村庄到其偏好池塘的最短路径数量相等？
输入的第一行包含两个整数 $n$ 和 $k$（$3 lt.eq n lt.eq 99$，$0 lt.eq k lt.eq 2 times (n -2)$），其中 $n$ 为城市的宽度（奇数），$k$ 为要放置的酒店数量。
如果存在一种放置所有酒店的方式满足题意，则输出 "_YES_"，否则输出 "_NO_"。
如果可能，请额外输出 4 行，描述城市布局，每行包含 $n$ 个字符，若该单元格有酒店则为 "_#_"，否则为 "_._"。
## Input
输入的第一行包含两个整数 $n$ 和 $k$（$3 lt.eq n lt.eq 99$，$0 lt.eq k lt.eq 2 times (n -2)$），其中 $n$ 为城市的宽度（奇数），$k$ 为要放置的酒店数量。
## Output
如果存在一种放置所有酒店的方式满足题意，则输出 "_YES_"，否则输出 "_NO_"。如果可能，请额外输出 4 行，描述城市布局，每行包含 $n$ 个字符，若该单元格有酒店则为 "_#_"，否则为 "_._"。
[samples]
**Definitions**
Let $ n \in \mathbb{Z} $ be an odd integer with $ 3 \leq n \leq 99 $, representing the number of columns in a $ 4 \times n $ grid.
Let $ k \in \mathbb{Z} $ with $ 0 \leq k \leq 2(n - 2) $, representing the number of hotels to place.
The grid has rows $ \{1, 2, 3, 4\} $ and columns $ \{1, \dots, n\} $.
Border cells are those with row $ \in \{1, 4\} $ or column $ \in \{1, n\} $.
Internal (non-border) cells are those with row $ \in \{2, 3\} $ and column $ \in \{2, \dots, n-1\} $.
There are $ 2(n - 2) $ internal cells available for hotel placement.
Two source-target pairs:
- Village A: $ (1, 1) \to $ Pond A: $ (4, n) $
- Village B: $ (4, 1) \to $ Pond B: $ (1, n) $
A path is a sequence of adjacent (sharing a side) non-hotel cells.
A shortest path moves only right and down (for A) or only right and up (for B).
Let $ P_A(S) $ denote the number of shortest paths from $ (1,1) $ to $ (4,n) $ in grid $ S $,
and $ P_B(S) $ denote the number of shortest paths from $ (4,1) $ to $ (1,n) $ in grid $ S $,
where $ S \subseteq \text{internal cells} $ is the set of cells occupied by hotels.
**Constraints**
1. $ n $ is odd, $ 3 \leq n \leq 99 $
2. $ 0 \leq k \leq 2(n - 2) $
3. Hotels are placed only in internal cells: $ S \subseteq \{2,3\} \times \{2, \dots, n-1\} $, $ |S| = k $
**Objective**
Determine whether there exists a set $ S $ of $ k $ internal cells such that:
$$
P_A(S) = P_B(S)
$$
If yes, output "YES" and a $ 4 \times n $ grid representation with `#` for hotels and `.` otherwise.
If no, output "NO".