H. Spinarak Symmetry
time limit per test
1 second
memory limit per test
256 megabytes
input
standard input
output
standard output

Hermione is a world-renowned Pokémon stylist who prefers to emphasize a Pokémon's abilities in their outfit. Her latest idea is a snazzy Spinarak-silk symmetric shawl and she needs your help to bring it into reality.

Hermione's workshop has a weaving station that can weave Spinarak silk into rectangular patterns of tiles. Each tile in the pattern can either have a hole in the middle – like lace – or be fully filled. The weaving station is automated and takes a specific format of ASCII input. The input begins with a declaration of the height $$$h$$$ and width $$$w$$$ of the pattern $$$(1 \le h, w, \le 100)$$$, and then $$$h$$$ lines of $$$w$$$-length strings consisting of '.' and '#' representing holed and filled tiles, respectively.

As a highly experienced stylist, Hermione has a very particular vision for this shawl, such that the design must obey the following rules:

  • The pattern must be both horizontally and vertically symmetrical.
  • There must be exactly $$$n$$$ filled tiles in the pattern.
  • There must be exactly $$$k$$$ areas of filled tiles in the pattern, where an "area" is a cardinally contiguous set of filled tiles.
  • The sizes of any two areas must not differ by more than $$$3$$$ – where an area's "size" is the number of filled tiles of which it consists.

Hermione has developed these constraints, but is unsure which combination of $$$n$$$ and $$$k$$$ will look best. She needs your help, so she will send over various combinations of $$$n$$$ and $$$k$$$ to prototype, and she expects you to send back an input pattern that matches all of the above constraints.

To ease your development, Hermione will only send combinations of $$$n$$$ and $$$k$$$ that have the same parity i.e. $$$n$$$ is odd if and only if $$$k$$$ is odd. It is guaranteed that a valid design always exists under these constraints.

Input

The first line contains two integers, $$$n$$$ and $$$k$$$ $$$(0 \lt k \le n \le 10^3, \; n \equiv k \text{ mod } 2)$$$, the number of contiguous areas and the number of filled tiles to include in your design, respectively.

Output

On the first line, output two integers, $$$h$$$ and $$$w$$$ $$$(0 \lt h, w \le 100)$$$, the dimensions of your design. On the next $$$h$$$ lines, print your design using $$$w$$$-length strings consisting of '.' for holed tiles and '#' for filled tiles.

Examples
Input
7 5
Output
1 11
#.#.###.#.#
Input
6 4
Output
6 5
.....
..#..
.#.#.
.#.#.
..#..
.....
Input
25 7
Output
9 11
...........
...##.##...
...##.##...
.....#.....
.##.###.##.
.....#.....
...##.##...
...##.##...
...........
Input
20 8
Output
12 12
............
............
..##....##..
..##....##..
....#..#....
............
............
....#..#....
..##....##..
..##....##..
............
............