Problem Statement

In the calendar of AtCoderLand, a year consists of M months: month 1, month 2, \dots, month M. The i-th month consists of D_i days: day 1, day 2, \dots, day D_i.
Furthermore, the number of days in a year is odd, that is, D_1+D_2+\dots+D_M is odd.
Find what day of what month is the middle day of the year.
In other words, let day 1 of month 1 be the first day, and find a and b such that the ((D_1+D_2+\dots+D_M+1)/2)-th day is day b of month a.

Constraints

• All input values are integers.
• 1 \le M \le 100
• 1 \le D_i \le 100
• D_1 + D_2 + \dots + D_M is odd.

Sample Input 1

12
31 28 31 30 31 30 31 31 30 31 30 31

Sample Output 1

7 2

In this input, a year consists of 31+28+31+30+31+30+31+31+30+31+30+31=365 days.
Let us find the middle day, which is the ((365+1)/2 = 183)-th day.

• Months 1,2,3,4,5,6 contain a total of 181 days.
• Day 1 of month 7 is the 182-th day.
• Day 2 of month 7 is the 183-th day.

Thus, the answer is day 2 of month 7.

Sample Input 2

1
1

Sample Output 2

1 1

Sample Input 3

6
3 1 4 1 5 9

Sample Output 3

5 3

Problem Statement

You are given a positive integer N. If there are integers x and y such that N=2^x3^y, print Yes; otherwise, print No.

Constraints

• 1\leq N\leq10^{18}
• N is an integer.

Sample Input 1

324

Sample Output 1

Yes

For x=2,y=4, we have 2^x3^y=2^23^4=4\times81=324, so the condition is satisfied. Thus, you should print Yes.

Sample Input 2

5

Sample Output 2

No

There are no integers x and y such that 2^x3^y=5. Thus, you should print No.

Sample Input 3

32

Sample Output 3

Yes

For x=5,y=0, we have 2^x3^y=32\times1=32, so you should print Yes.

Sample Input 4

37748736

Sample Output 4

Yes

Problem Statement

You are given two strings S and T. Determine whether it is possible to make S equal T by performing the following operation some number of times (possibly zero).

Between two consecutive equal characters in S, insert a character equal to these characters. That is, take the following three steps.

1. Let N be the current length of S, and S = S_1S_2\ldots S_N.
2. Choose an integer i between 1 and N-1 (inclusive) such that S_i = S_{i+1}. (If there is no such i, do nothing and terminate the operation now, skipping step 3.)
3. Insert a single copy of the character S_i(= S_{i+1}) between the i-th and (i+1)-th characters of S. Now, S is a string of length N+1: S_1S_2\ldots S_i S_i S_{i+1} \ldots S_N.

Constraints

• Each of S and T is a string of length between 2 and 2 \times 10^5 (inclusive) consisting of lowercase English letters.

Sample Input 1

abbaac
abbbbaaac

Sample Output 1

Yes

You can make S = abbaac equal T = abbbbaaac by the following three operations.

• First, insert b between the 2-nd and 3-rd characters of S. Now, S = abbbaac.
• Next, insert b again between the 2-nd and 3-rd characters of S. Now, S = abbbbaac.
• Lastly, insert a between the 6-th and 7-th characters of S. Now, S = abbbbaaac.

Thus, Yes should be printed.

Sample Input 2

xyzz
xyyzz

Sample Output 2

No

No sequence of operations makes S = xyzz equal T = xyyzz. Thus, No should be printed.

Problem Statement

Takahashi has two sheets A and B, each composed of black squares and transparent squares, and an infinitely large sheet C composed of transparent squares.
There is also an ideal sheet X for Takahashi composed of black squares and transparent squares.

The sizes of sheets A, B, and X are H_A rows \times W_A columns, H_B rows \times W_B columns, and H_X rows \times W_X columns, respectively.
The squares of sheet A are represented by H_A strings of length W_A, A_1, A_2, \ldots, A_{H_A} consisting of . and #.
If the j-th character (1\leq j\leq W_A) of A_i (1\leq i\leq H_A) is ., the square at the i-th row from the top and j-th column from the left is transparent; if it is #, that square is black.
Similarly, the squares of sheets B and X are represented by H_B strings of length W_B, B_1, B_2, \ldots, B_{H_B}, and H_X strings of length W_X, X_1, X_2, \ldots, X_{H_X}, respectively.

Takahashi's goal is to create sheet X using all black squares in sheets A and B by following the steps below with sheets A, B, and C.

1. Paste sheets A and B onto sheet C along the grid. Each sheet can be pasted anywhere by translating it, but it cannot be cut or rotated.
2. Cut out an H_X\times W_X area from sheet C along the grid. Here, a square of the cut-out sheet will be black if a black square of sheet A or B is pasted there, and transparent otherwise.

Determine whether Takahashi can achieve his goal by appropriately choosing the positions where the sheets are pasted and the area to cut out, that is, whether he can satisfy both of the following conditions.

• The cut-out sheet includes all black squares of sheets A and B. The black squares of sheets A and B may overlap on the cut-out sheet.
• The cut-out sheet coincides sheet X without rotating or flipping.

Constraints

• 1\leq H_A, W_A, H_B, W_B, H_X, W_X\leq 10
• H_A, W_A, H_B, W_B, H_X, W_X are integers.
• A_i is a string of length W_A consisting of . and #.
• B_i is a string of length W_B consisting of . and #.
• X_i is a string of length W_X consisting of . and #.
• Sheets A, B, and X each contain at least one black square.

Sample Input 1

3 5
#.#..
.....
.#...
2 2
#.
.#
5 3
...
#.#
.#.
.#.
...

Sample Output 1

Yes

First, paste sheet A onto sheet C, as shown in the figure below.

     \vdots
  .......  
  .#.#...  
\cdots.......\cdots
  ..#....  
  .......  
     \vdots

Next, paste sheet B so that its top-left corner aligns with that of sheet A, as shown in the figure below.

     \vdots
  .......  
  .#.#...  
\cdots..#....\cdots
  ..#....  
  .......  
     \vdots

Now, cut out a 5\times 3 area with the square in the first row and second column of the range illustrated above as the top-left corner, as shown in the figure below.

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

This includes all black squares of sheets A and B and matches sheet X, satisfying the conditions.

Therefore, print Yes.

Sample Input 2

2 2
#.
.#
2 2
#.
.#
2 2
##
##

Sample Output 2

No

Note that sheets A and B may not be rotated or flipped when pasting them.

Sample Input 3

1 1
#
1 2
##
1 1
#

Sample Output 3

No

No matter how you paste or cut, you cannot cut out a sheet that includes all black squares of sheet B, so you cannot satisfy the first condition. Therefore, print No.

Sample Input 4

3 3
###
...
...
3 3
#..
#..
#..
3 3
..#
..#
###

Sample Output 4

Yes

Problem Statement

The Republic of AtCoder lies on a Cartesian coordinate plane.
It has N towns, numbered 1,2,\dots,N. Town i is at (x_i, y_i), and no two different towns are at the same coordinates.

There are teleportation spells in the nation. A spell is identified by a pair of integers (a,b), and casting a spell (a, b) at coordinates (x, y) teleports you to (x+a, y+b).

Snuke is a great magician who can learn the spell (a, b) for any pair of integers (a, b) of his choice. The number of spells he can learn is also unlimited.
To be able to travel between the towns using spells, he has decided to learn some number of spells so that it is possible to do the following for every pair of different towns (i, j).

• Choose just one spell among the spells learned. Then, repeatedly use just the chosen spell to get from Town i to Town j.

At least how many spells does Snuke need to learn to achieve the objective above?

Constraints

• 2 \leq N \leq 500
• 0 \leq x_i \leq 10^9 (1 \leq i \leq N)
• 0 \leq y_i \leq 10^9 (1 \leq i \leq N)
• (x_i, y_i) \neq (x_j, y_j) if i \neq j.

Sample Input 1

3
1 2
3 6
7 4

Sample Output 1

6

The figure below illustrates the positions of the towns (along with the coordinates of the four corners).

If Snuke learns the six spells below, he can get from Town i to Town j by using one of the spells once for every pair (i,j) (i \neq j), achieving his objective.

• (2, 4)
• (-2, -4)
• (4, -2)
• (-4, 2)
• (-6, -2)
• (6, 2)

Another option is to learn the six spells below. In this case, he can get from Town i to Town j by using one of the spells twice for every pair (i,j) (i \neq j), achieving his objective.

• (1, 2)
• (-1, -2)
• (2, -1)
• (-2, 1)
• (-3, -1)
• (3, 1)

There is no combination of spells that consists of less than six spells and achieve the objective, so we should print 6.

Sample Input 2

3
1 2
2 2
4 2

Sample Output 2

2

The optimal choice is to learn the two spells below:

• (1, 0)
• (-1, 0)

Sample Input 3

4
0 0
0 1000000000
1000000000 0
1000000000 1000000000

Sample Output 3

8