Problem Statement

Takahashi runs an SNS "Twidai," which has N users from user 1 through user N. In Twidai, users can follow or unfollow other users.

Q operations have been performed since Twidai was launched. The i-th (1\leq i\leq Q) operation is represented by three integers T_i, A_i, and B_i, whose meanings are as follows:

• If T_i = 1: it means that user A_i follows user B_i. If user A_i is already following user B_i at the time of this operation, it does not make any change.
• If T_i = 2: it means that user A_i unfollows user B_i. If user A_i is not following user B_i at the time of this operation, it does not make any change.
• If T_i = 3: it means that you are asked to determine if users A_i and B_i are following each other. You need to print Yes if user A_i is following user B_i and user B_i is following user A_i, and No otherwise.

When the service was launched, no users were following any users.

Print the correct answers for all operations such that T_i = 3 in ascending order of i.

Constraints

• 2 \leq N \leq 10 ^ 9
• 1 \leq Q \leq 2\times10 ^ 5
• T _ i=1,2,3\ (1\leq i\leq Q)
• 1 \leq A _ i \leq N\ (1\leq i\leq Q)
• 1 \leq B _ i \leq N\ (1\leq i\leq Q)
• A _ i\neq B _ i\ (1\leq i\leq Q)
• There exists i\ (1\leq i\leq Q) such that T _ i=3.
• All values in the input are integers.

Sample Input 1

3 9
1 1 2
3 1 2
1 2 1
3 1 2
1 2 3
1 3 2
3 1 3
2 1 2
3 1 2

Sample Output 1

No
Yes
No
No

Twidai has three users. The nine operations are as follows.

• User 1 follows user 2. No other users are following or followed by any users.
• Determine if users 1 and 2 are following each other. User 1 is following user 2, but user 2 is not following user 1, so No is the correct answer to this operation.
• User 2 follows user 1.
• Determine if users 1 and 2 are following each other. User 1 is following user 2, and user 2 is following user 1, so Yes is the correct answer to this operation.
• User 2 follows user 3.
• User 3 follows user 2.
• Determine if users 1 and 3 are following each other. User 1 is not following user 3, and user 3 is not following user 1, so No is the correct answer to this operation.
• User 1 unfollows user 2.
• Determine if users 1 and 2 are following each other. User 2 is following user 1, but user 1 is not following user 2, so No is the correct answer to this operation.

Sample Input 2

2 8
1 1 2
1 2 1
3 1 2
1 1 2
1 1 2
1 1 2
2 1 2
3 1 2

Sample Output 2

Yes
No

A user may follow the same user multiple times.

Sample Input 3

10 30
3 1 6
3 5 4
1 6 1
3 1 7
3 8 4
1 1 6
2 4 3
1 6 5
1 5 6
1 1 8
1 8 1
2 3 10
1 7 6
3 5 6
1 6 7
3 6 7
1 9 5
3 8 6
3 3 8
2 6 9
1 7 1
3 10 8
2 9 2
1 10 9
2 6 10
2 6 8
3 1 6
3 1 8
2 8 5
1 9 10

Sample Output 3

No
No
No
No
Yes
Yes
No
No
No
Yes
Yes

Problem Statement

We have a grid with H horizontal rows and W vertical columns. (i, j) denotes the square at the i-th row from the top and j-th column from the left.
(i,j) has a character G_{i,j} written on it. G_{i,j} is U, D, L, or R.

You are initially at (1,1). You repeat the following operation until you cannot make a move.

Let (i,j) be the square you are currently at.
If G_{i,j} is U and i \neq 1, move to (i-1,j).
If G_{i,j} is D and i \neq H, move to (i+1,j).
If G_{i,j} is L and j \neq 1, move to (i,j-1).
If G_{i,j} is R and j \neq W, move to (i,j+1).
Otherwise, you cannot make a move.

Print the square you end up at when you cannot make a move.
If you indefinitely repeat moving, print -1 instead.

Constraints

• 1 \leq H, W \leq 500
• G_{i,j} is U, D, L, or R.
• H and W are integers.

Sample Input 1

2 3
RDU
LRU

Sample Output 1

1 3

You will move as (1, 1) \to (1, 2) \to (2, 2) \to (2, 3) \to (1, 3), ending up here, so the answer is (1, 3).

Sample Input 2

2 3
RRD
ULL

Sample Output 2

-1

You will indefinitely repeat moving as (1, 1) \to (1, 2) \to (1, 3) \to (2, 3) \to (2, 2) \to (2, 1) \to (1, 1) \to (1, 2) \to \dots, so -1 should be printed in this case.

Sample Input 3

9 44
RRDDDDRRRDDDRRRRRRDDDRDDDDRDDRDDDDDDRRDRRRRR
RRRDLRDRDLLLLRDRRLLLDDRDLLLRDDDLLLDRRLLLLLDD
DRDLRLDRDLRDRLDRLRDDLDDLRDRLDRLDDRLRRLRRRDRR
DDLRRDLDDLDDRLDDLDRDDRDDDDRLRRLRDDRRRLDRDRDD
RDLRRDLRDLLLLRRDLRDRRDRRRDLRDDLLLLDDDLLLLRDR
RDLLLLLRDLRDRLDDLDDRDRRDRLDRRRLDDDLDDDRDDLDR
RDLRRDLDDLRDRLRDLDDDLDDRLDRDRDLDRDLDDLRRDLRR
RDLDRRLDRLLLLDRDRLLLRDDLLLLLRDRLLLRRRRLLLDDR
RRRRDRDDRRRDDRDDDRRRDRDRDRDRRRRRRDDDRDDDDRRR

Sample Output 3

9 5

Problem Statement

You are given a sequence of positive integers: A=(a_1,a_2,\ldots,a_N).
You can choose and perform one of the following operations any number of times, possibly zero.

• Choose an integer i such that 1 \leq i \leq N and a_i is a multiple of 2, and replace a_i with \frac{a_i}{2}.
• Choose an integer i such that 1 \leq i \leq N and a_i is a multiple of 3, and replace a_i with \frac{a_i}{3}.

Your objective is to make A satisfy a_1=a_2=\ldots=a_N.
Find the minimum total number of times you need to perform an operation to achieve the objective. If there is no way to achieve the objective, print -1 instead.

Constraints

• 2 \leq N \leq 1000
• 1 \leq a_i \leq 10^9
• All values in the input are integers.

Sample Input 1

3
1 4 3

Sample Output 1

3

Here is a way to achieve the objective in three operations, which is the minimum needed.

• Choose an integer i=2 such that a_i is a multiple of 2, and replace a_2 with \frac{a_2}{2}. A becomes (1,2,3).
• Choose an integer i=2 such that a_i is a multiple of 2, and replace a_2 with \frac{a_2}{2}. A becomes (1,1,3).
• Choose an integer i=3 such that a_i is a multiple of 3, and replace a_3 with \frac{a_3}{3}. A becomes (1,1,1).

Sample Input 2

3
2 7 6

Sample Output 2

-1

There is no way to achieve the objective.

Sample Input 3

6
1 1 1 1 1 1

Sample Output 3

0

Problem Statement

You are given a sequence of N numbers: A=(A_1,A_2,\ldots,A_N). Here, each A_i (1\leq i\leq N) satisfies 1\leq A_i \leq N.

Takahashi and Aoki will play N rounds of a game. For each i=1,2,\ldots,N, the i-th game will be played as follows.

1. Aoki specifies a positive integer K_i.
2. After knowing K_i Aoki has specified, Takahashi chooses an integer S_i between 1 and N, inclusive, and writes it on a blackboard.

3. Repeat the following K_i times.

   • Replace the integer x written on the blackboard with A_x.

If i is written on the blackboard after the K_i iterations, Takahashi wins the i-th round; otherwise, Aoki wins.
Here, K_i and S_i can be chosen independently for each i=1,2,\ldots,N.

Find the number of rounds Takahashi wins if both players play optimally to win.

Constraints

• 1\leq N\leq 2\times 10^5
• 1\leq A_i\leq N
• All values in the input are integers.

Sample Input 1

3
2 2 3

Sample Output 1

2

In the first round, if Aoki specifies K_1=2, Takahashi cannot win whichever option he chooses for S_1: 1, 2, or 3.

For example, if Takahashi writes S_1=1 on the initial blackboard, the two operations will change this number as follows: 1\to 2(=A_1), 2\to 2(=A_2). The final number on the blackboard will be 2(\neq 1), so Aoki wins.

On the other hand, in the second and third rounds, Takahashi can win by writing 2 and 3, respectively, on the initial blackboard, whatever value Aoki specifies as K_i.

Therefore, if both players play optimally to win, Takashi wins two rounds: the second and the third. Thus, you should print 2.

Sample Input 2

2
2 1

Sample Output 2

2

In the first round, Takahashi can win by writing 2 on the initial blackboard if K_1 specified by Aoki is odd, and 1 if it is even.

Similarly, there is a way for Takahashi to win the second round. Thus, Takahashi can win both rounds: the answer is 2.

Problem Statement

Given is a string S consisting of digits from 1 through 9.
From this string S, let us make a formula T by the following operations.

• Initially, let T=S.
• Choose a (possibly empty) set A of different integers where each element is between 1 and |S|-1 (inclusive).
• For each element x in descending order, do the following.
  • Insert a + between the x-th and (x+1)-th characters of T.

For example, when S= 1234 and A= \lbrace 2,3 \rbrace, we will have T= 12+3+4.

Consider evaluating all possible formulae T obtained by these operations. Find the sum, modulo 998244353, of the evaluations.

Constraints

• 1 \le |S| \le 2 \times 10^5
• S consists of 1, 2, 3, 4, 5, 6, 7, 8, and 9.

Sample Input 1

1234

Sample Output 1

1736

There are eight formulae that can be obtained as T: 1234, 123+4, 12+34, 12+3+4, 1+234, 1+23+4, 1+2+34, and 1+2+3+4.
The sum of the evaluations of these formulae is 1736.

Sample Input 2

1

Sample Output 2

1

S may have a length of 1, in which case the only possible choice for A is the empty set.

Sample Input 3

31415926535897932384626433832795

Sample Output 3

85607943

Be sure to find the sum modulo 998244353.