Question:
Monocarp has got an array a consisting of n integers. Let's denote k as the mathematic mean of these elements (note that it's possible that k is not an integer).

The mathematic mean of an array of n elements is the sum of elements divided by the number of these elements (i. e. sum divided by n).

Monocarp wants to delete exactly two elements from a so that the mathematic mean of the remaining (n - 2) elements is still equal to k.

Your task is to calculate the number of pairs of positions [i, j] (i < j) such that if the elements on these positions are deleted, the mathematic mean of (n - 2) remaining elements is equal to k (that is, it is equal to the mathematic mean of n elements of the original array a).

Input

The first line contains a single integer t (1 ≤ t ≤ 10^4) — the number of testcases.

The first line of each testcase contains one integer n (3 ≤ n ≤ 2 ⋅ 10^5) — the number of elements in the array.

The second line contains a sequence of integers a_1, a_2, ..., a_n (0 ≤ a_i ≤ 10^{9}), where a_i is the i-th element of the array.

The sum of n over all testcases doesn't exceed 2 ⋅ 10^5.

Output

Print one integer — the number of pairs of positions [i, j] (i < j) such that if the elements on these positions are deleted, the mathematic mean of (n - 2) remaining elements is equal to k (that is, it is equal to the mathematic mean of n elements of the original array a).

Example

Input

4
4
8 8 8 8
3
50 20 10
5
1 4 7 3 5
7
1 2 3 4 5 6 7

Output

6
0
2
3

Note

In the first example, any pair of elements can be removed since all of them are equal.

In the second example, there is no way to delete two elements so the mathematic mean doesn't change.

In the third example, it is possible to delete the elements on positions 1 and 3, or the elements on positions 4 and 5.


Question:
Monocarp is the coach of the Berland State University programming teams. He decided to compose a problemset for a training session for his teams.

Monocarp has n problems that none of his students have seen yet. The i-th problem has a topic a_i (an integer from 1 to n) and a difficulty b_i (an integer from 1 to n). All problems are different, that is, there are no two tasks that have the same topic and difficulty at the same time.

Monocarp decided to select exactly 3 problems from n problems for the problemset. The problems should satisfy at least one of two conditions (possibly, both):

  * the topics of all three selected problems are different;
  * the difficulties of all three selected problems are different.

Your task is to determine the number of ways to select three problems for the problemset.

Input

The first line contains a single integer t (1 ≤ t ≤ 50000) — the number of testcases.

The first line of each testcase contains an integer n (3 ≤ n ≤ 2 ⋅ 10^5) — the number of problems that Monocarp have.

In the i-th of the following n lines, there are two integers a_i and b_i (1 ≤ a_i, b_i ≤ n) — the topic and the difficulty of the i-th problem.

It is guaranteed that there are no two problems that have the same topic and difficulty at the same time.

The sum of n over all testcases doesn't exceed 2 ⋅ 10^5.

Output

Print the number of ways to select three training problems that meet either of the requirements described in the statement.

Example

Input

2
4
2 4
3 4
2 1
1 3
5
1 5
2 4
3 3
4 2
5 1

Output

3
10

Note

In the first example, you can take the following sets of three problems:

  * problems 1, 2, 4;
  * problems 1, 3, 4;
  * problems 2, 3, 4.

Thus, the number of ways is equal to three.


Question:
You are given a matrix, consisting of n rows and m columns. The rows are numbered top to bottom, the columns are numbered left to right.

Each cell of the matrix can be either free or locked.

Let's call a path in the matrix a staircase if it:

  * starts and ends in the free cell;
  * visits only free cells;
  * has one of the two following structures:
    1. the second cell is 1 to the right from the first one, the third cell is 1 to the bottom from the second one, the fourth cell is 1 to the right from the third one, and so on;
    2. the second cell is 1 to the bottom from the first one, the third cell is 1 to the right from the second one, the fourth cell is 1 to the bottom from the third one, and so on.

In particular, a path, consisting of a single cell, is considered to be a staircase.

Here are some examples of staircases:

<image>

Initially all the cells of the matrix are free.

You have to process q queries, each of them flips the state of a single cell. So, if a cell is currently free, it makes it locked, and if a cell is currently locked, it makes it free.

Print the number of different staircases after each query. Two staircases are considered different if there exists such a cell that appears in one path and doesn't appear in the other path.

Input

The first line contains three integers n, m and q (1 ≤ n, m ≤ 1000; 1 ≤ q ≤ 10^4) — the sizes of the matrix and the number of queries.

Each of the next q lines contains two integers x and y (1 ≤ x ≤ n; 1 ≤ y ≤ m) — the description of each query.

Output

Print q integers — the i-th value should be equal to the number of different staircases after i queries. Two staircases are considered different if there exists such a cell that appears in one path and doesn't appear in the other path.

Examples

Input

2 2 8
1 1
1 1
1 1
2 2
1 1
1 2
2 1
1 1

Output

5
10
5
2
5
3
1
0

Input

3 4 10
1 4
1 2
2 3
1 2
2 3
3 2
1 3
3 4
1 3
3 1

Output

49
35
24
29
49
39
31
23
29
27

Input

1000 1000 2
239 634
239 634

Output

1332632508
1333333000


Question:
A bracket sequence is a string containing only characters "(" and ")". A regular bracket sequence (or, shortly, an RBS) is a bracket sequence that can be transformed into a correct arithmetic expression by inserting characters "1" and "+" between the original characters of the sequence. For example:

  * bracket sequences "()()" and "(())" are regular (the resulting expressions are: "(1)+(1)" and "((1+1)+1)");
  * bracket sequences ")(", "(" and ")" are not.

Let's denote the concatenation of two strings x and y as x+y. For example, "()()" + ")(" = "()())(".

You are given n bracket sequences s_1, s_2, ..., s_n. You can rearrange them in any order (you can rearrange only the strings themselves, but not the characters in them).

Your task is to rearrange the strings in such a way that the string s_1 + s_2 + ... + s_n has as many non-empty prefixes that are RBS as possible.

Input

The first line contains a single integer n (1 ≤ n ≤ 20).

Then n lines follow, the i-th of them contains s_i — a bracket sequence (a string consisting of characters "(" and/or ")". All sequences s_i are non-empty, their total length does not exceed 4 ⋅ 10^5.

Output

Print one integer — the maximum number of non-empty prefixes that are RBS for the string s_1 + s_2 + ... + s_n, if the strings s_1, s_2, ..., s_n can be rearranged arbitrarily.

Examples

Input

2
(
)

Output

1

Input

4
()()())
(
(
)

Output

4

Input

1
(())

Output

1

Input

1
)(()

Output

0

Note

In the first example, you can concatenate the strings as follows: "(" + ")" = "()", the resulting string will have one prefix, that is an RBS: "()".

In the second example, you can concatenate the strings as follows: "(" + ")" + "()()())" + "(" = "()()()())(", the resulting string will have four prefixes that are RBS: "()", "()()", "()()()", "()()()()".

The third and the fourth examples contain only one string each, so the order is fixed.


Question:
Let's call a positive integer good if there is no digit 0 in its decimal representation.

For an array of a good numbers a, one found out that the sum of some two neighboring elements is equal to x (i.e. x = a_i + a_{i + 1} for some i). x had turned out to be a good number as well.

Then the elements of the array a were written out one after another without separators into one string s. For example, if a = [12, 5, 6, 133], then s = 1256133.

You are given a string s and a number x. Your task is to determine the positions in the string that correspond to the adjacent elements of the array that have sum x. If there are several possible answers, you can print any of them.

Input

The first line contains the string s (2 ≤ |s| ≤ 5 ⋅ 10^5).

The second line contains an integer x (2 ≤ x < 10^{200000}).

An additional constraint on the input: the answer always exists, i.e you can always select two adjacent substrings of the string s so that if you convert these substrings to integers, their sum is equal to x.

Output

In the first line, print two integers l_1, r_1, meaning that the first term of the sum (a_i) is in the string s from position l_1 to position r_1.

In the second line, print two integers l_2, r_2, meaning that the second term of the sum (a_{i + 1}) is in the string s from position l_2 to position r_2.

Examples

Input

1256133
17

Output

1 2
3 3

Input

9544715561
525

Output

2 3
4 6

Input

239923
5

Output

1 1
2 2

Input

1218633757639
976272

Output

2 7
8 13

Note

In the first example s[1;2] = 12 and s[3;3] = 5, 12+5=17.

In the second example s[2;3] = 54 and s[4;6] = 471, 54+471=525.

In the third example s[1;1] = 2 and s[2;2] = 3, 2+3=5.

In the fourth example s[2;7] = 218633 and s[8;13] = 757639, 218633+757639=976272.


Question:
You are given an array A of length N weights of masses A_1, A_2...A_N. No two weights have the same mass. You can put every weight on one side of the balance (left or right). You don't have to put weights in order A_1,...,A_N. There is also a string S consisting of characters "L" and "R", meaning that after putting the i-th weight (not A_i, but i-th weight of your choice) left or right side of the balance should be heavier. Find the order of putting the weights on the balance such that rules of string S are satisfied.

Input

The first line contains one integer N (1 ≤ N ≤ 2*10^5) - the length of the array A The second line contains N distinct integers: A_1, A_2,...,A_N (1 ≤ A_i ≤ 10^9) - the weights given The third line contains string S of length N consisting only of letters "L" and "R" - string determining which side of the balance should be heavier after putting the i-th weight of your choice

Output

The output contains N lines. In every line, you should print one integer and one letter - integer representing the weight you are putting on the balance in that move and the letter representing the side of the balance where you are putting the weight. If there is no solution, print -1.

Example

Input

5
3 8 2 13 7
LLRLL

Output

3 L
2 R
8 R
13 L
7 L

Note

Explanation for the test case:

after the 1st weight: 3 L (left side is heavier)

after the 2nd weight: 2 R (left side is heavier)

after the 3rd weight: 8 R (right side is heavier)

after the 4th weight: 13 L (left side is heavier)

after the 5th weight: 7 L (left side is heavier)

So, the rules given by string S are fulfilled and our order of putting the weights is correct.


Question:
Alice and Bob always had hard time choosing restaurant for the dinner. Previously they performed Eenie Meenie Miney Mo game, but eventually as their restaurant list grew, they had to create a new game. This new game starts as they write restaurant names on N cards and align the cards in one line. Before the game begins, they both choose starting card and starting direction they are going to. They take turns in order one after another. After each turn, they move one card in their current direction. If they reach the end or beginning of the line of cards they change direction. Once they meet in a card, the card is marked for removal and is removed the first moment they both leave the card.

<image> Example of how card is removed

They repeat this process until there is only one restaurant card left. Since there are a lot of restaurant cards, they are bored to simulate this process over and over and need your help to determine the last card that remains. Can you help them?

Input

The first line of the input is one integer T (1 ≤ T ≤ 10^{4}) representing number of test cases. Each test case contains 3 lines: The first line contains an integer N representing initial number of cards. Next line contains two integer values A,B (0 ≤ A, B < N, 2 ≤ N ≤ 10^{18}) representing starting 0-based index of the card in the array. Last line contains two strings D_A, D_B ∈ {"left", "right"} representing starting direction of their movement.

Output

The output contains T integer number – the 0-based index of the last card that remains for every test case in order.

Example

Input

1
4
0 1
left right

Output

0

Note

Note that since Alice is starting at the beginning of the line even though her initial direction is left, on her next move she will go right.

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Question:
Little Johnny Bubbles enjoys spending hours in front of his computer playing video games. His favorite game is Bubble Strike, fast-paced bubble shooting online game for two players.

Each game is set in one of the N maps, each having different terrain configuration. First phase of each game decides on which map the game will be played. The game system randomly selects three maps and shows them to the players. Each player must pick one of those three maps to be discarded. The game system then randomly selects one of the maps that were not picked by any of the players and starts the game.

Johnny is deeply enthusiastic about the game and wants to spend some time studying maps, thus increasing chances to win games played on those maps. However, he also needs to do his homework, so he does not have time to study all the maps. That is why he asked himself the following question: "What is the minimum number of maps I have to study, so that the probability to play one of those maps is at least P"?

Can you help Johnny find the answer for this question? You can assume Johnny's opponents do not know him, and they will randomly pick maps.

Input

The first line contains two integers N (3 ≤ N ≤ 10^{3}) and P (0 ≤ P ≤ 1) – total number of maps in the game and probability to play map Johnny has studied. P will have at most four digits after the decimal point.

Output

Output contains one integer number – minimum number of maps Johnny has to study.

Example

Input

7 1.0000

Output

6


Question:
There are N bubbles in a coordinate plane. Bubbles are so tiny that it can be assumed that each bubble is a point (X_i, Y_i).

Q Bubble Cup finalists plan to play with the bubbles. Each finalist would link to use infinitely long Bubble Cup stick to pop some bubbles. The i-th finalist would like to place the stick in the direction of vector (dxi, dyi), and plays the following game until K_i bubbles are popped. The game starts with finalist placing the stick in the direction of vector (dx_i, dy_i), and sweeping it from the infinity to the left until it hits some bubble, which is immediately popped. It is guaranteed that only one bubble will be hit in this step. After that the finalist starts rotating the stick in the counter clockwise direction with the center of rotation in point where the previous bubble was popped. When the next bubble is hit, it is immediately popped and becomes the new center of rotation. The process continues until K_i bubbles have been popped. It is guaranteed that the stick won't hit two bubbles simultaneously in this process.

For each finalist find which bubble would be popped the last. Note that each game starts with the configuration of all N bubbles, so the games don't depend on the previous games.

Input

The first line contains one integer N — the number of bubbles. (1 ≤ N ≤ 10^5)

Each of the next N lines contains two integers. The i-th line contains integers X_i and Y_i — the coordinates of the i-th bubble. (-10^9 ≤ X_i, Y_i ≤ 10^9, (X_i, Y_i) ≠ (X_j, Y_j) for i ≠ j)

The next line contains one integer Q — the number of finalists willing to play with the bubbles. (1 ≤ Q ≤ 10^5)

Each of the next Q lines contains 3 integers. The i-th line contains integers dx_i, dy_i and K_i. (-10^9 ≤ dx_i, dy_i ≤ 10^9, 1 ≤ K_i ≤ N)

Output

For each of the Q finalists, print the index of the bubble which would be popped last, in the separate line.

Examples

Input

4
0 0
1 0
0 1
1 1
2
1 -1 3
-1 1 4

Output

4
2

Input

4
1 1
2 2
7 1
1 7
3
2 2 1
1 -5 4
-6 5 3

Output

3
2
3

Note

There are two finalists willing to play with the bubbles. If the first finalist plays with the bubbles, then the bubbles at coordinates (0, 0), (1, 0) and (1, 1) would be popped in that order. Their indexes are 1, 2 and 4, so the answer is 4. If the second finalist plays with the bubbles, then the bubbles at coordinates (1, 1), (0, 1), (0, 0) and (1, 0) would be popped in that order, so the answer is 2.

Visualization: [link](https://petljamediastorage.blob.core.windows.net/uploads/example1.gif).


Question:
You are given two integer arrays of length N, A1 and A2. You are also given Q queries of 4 types:

1 k l r x: set Ak_i:=min(Ak_i, x) for each l ≤ i ≤ r.

2 k l r x: set Ak_i:=max(Ak_i, x) for each l ≤ i ≤ r.

3 k l r x: set Ak_i:=Ak_i+x for each l ≤ i ≤ r.

4 l r: find the (∑_{i=l}^r F(A1_i+A2_i)) \% (10^9+7) where F(k) is the k-th Fibonacci number (F(0)=0, F(1)=1, F(k)=F(k-1)+F(k-2)), and x \% y denotes the remainder of the division of x by y.

You should process these queries and answer each query of the fourth type.

Input

The first line contains two integers N and Q. (1 ≤ N, Q ≤ 5 × 10^4)

The second line contains N integers, array A1_1, A1_2, ... A1_N. (0 ≤ A1_i ≤ 10^6)

The third line contains N integers, array A2_1, A2_2, ... A2_N. (0 ≤ A2_i ≤ 10^6)

The next Q lines describe the queries. Each line contains 5 or 3 integers, where the first integer denotes the type of the query. (k ∈ \{1, 2\}, 1 ≤ l ≤ r ≤ N)

For queries of type 1 and 2, 0 ≤ x ≤ 10^9 holds.

For queries of type 3, −10^6 ≤ x ≤ 10^6 holds.

It is guaranteed that after every query each number in arrays A1 and A2 will be nonnegative.

Output

Print the answer to each query of the fourth type, in separate lines.

Examples

Input

3 4
1 0 2
2 1 0
4 1 3
3 2 2 2 3
1 1 1 3 0
4 1 3

Output

4
4

Input

5 4
1 3 5 3 2
4 2 1 3 3
4 1 3
4 2 5
2 1 2 4 6
4 2 4

Output

18
26
68

Note

In the first example: The answer for the first query is F(1 + 2) + F(0 + 1) + F(2 + 0) = F(3) + F(1) + F(2) = 2 + 1 + 1 = 4. After the second query, the array A2 changes to [2, 4, 0]. After the third query, the array A1 changes to [0, 0, 0]. The answer for the fourth query is F(0 + 2) + F(0 + 4) + F(0 + 0) = F(2) + F(4) + F(0) = 1 + 3 + 0 = 4.

In the second example: The answer for the first query is F(1 + 4) + F(3 + 2) + F(5 + 1) = F(5) + F(5) + F(6) = 5 + 5 + 8 = 18. The answer for the second query is F(3 + 2) + F(5 + 1) + F(3 + 3) + F(2 + 3) = F(5) + F(6) + F(6) + F(5) = 5 + 8 + 8 + 5 = 26. After the third query, the array A1 changes to [1, 6, 6, 6, 2]. The answer for the fourth query is F(6 + 2) + F(6 + 1) + F(6 + 3) = F(8) + F(7) + F(9) = 21 + 13 + 34 = 68.


Question:
In the year 2420 humans have finally built a colony on Mars thanks to the work of Elon Tusk. There are 10^9+7 cities arranged in a circle in this colony and none of them are connected yet. Elon Tusk wants to connect some of those cities using only roads of the same size in order to lower the production cost of those roads. Because of that he gave a list on N cites where some cites can appear more than once and Q queries that you need to answer. For the query you need to determine if it is possible to connect all the cities from L_{i} to R_{i} on that list using only roads of length D_{i}.

Input

The first line contains two integers N and Q (1 ≤ N, Q ≤ 2⋅10^5 ) — the length of the array of cities and the number of queries you need to answer.

The second lines contains N integers representing the array of cites. Next Q lines contain three integers L, R and D (1 ≤ L_{i}, R_{i} ≤ N, 0 ≤ D_{i} ≤ 10^9+6) — the range of cities that needs to be connected and the length of the road that you can use.

Output

The output contains Q lines. If it is possible to connect all the cities from the i-th query can be connected with roads of length D_{i} the i-th line should contain the word "Yes", otherwise it should contain the word "No".

Examples

Input

9 8
17 0 12 6 10 8 2 4 5
2 3 12
2 3 6
2 4 6
4 6 2
2 8 2
1 2 17
1 8 2
9 9 14

Output

Yes
No
Yes
Yes
Yes
Yes
No
Yes

Input

4 1
7 21 14 0
1 4 1000000000

Output

Yes

Note

In the 5^{th} query of the first test case we can connect cities in this order 0-2-4-6-8-10-12 this way distance between any two connected cities is 2. In the second test case we can connect cities in this order 21-14-7-0 this way distance between any two connected cities is 10^9 module 10^9+7.


Question:
You are given N points on an infinite plane with the Cartesian coordinate system on it. N-1 points lay on one line, and one point isn't on that line. You are on point K at the start, and the goal is to visit every point. You can move between any two points in a straight line, and you can revisit points. What is the minimum length of the path?

Input

The first line contains two integers: N (3 ≤ N ≤ 2*10^5) - the number of points, and K (1 ≤ K ≤ N) - the index of the starting point.

Each of the next N lines contain two integers, A_i, B_i (-10^6 ≤ A_i, B_i ≤ 10^6) - coordinates of the i-th point.

Output

The output contains one number - the shortest path to visit all given points starting from point K. The absolute difference between your solution and the main solution shouldn't exceed 10^-6;

Example

Input

5 2
0 0
-1 1
2 -2
0 1
-2 2

Output

7.478709

Note

The shortest path consists of these moves:

2 -> 5

5 -> 4

4 -> 1

1 -> 3

There isn't any shorter path possible.


Question:
This is an interactive problem!

As part of your contribution in the Great Bubble War, you have been tasked with finding the newly built enemy fortress. The world you live in is a giant 10^9 × 10^9 grid, with squares having both coordinates between 1 and 10^9.

You know that the enemy base has the shape of a rectangle, with the sides parallel to the sides of the grid. The people of your world are extremely scared of being at the edge of the world, so you know that the base doesn't contain any of the squares on the edges of the grid (the x or y coordinate being 1 or 10^9).

To help you locate the base, you have been given a device that you can place in any square of the grid, and it will tell you the manhattan distance to the closest square of the base. The manhattan distance from square (a, b) to square (p, q) is calculated as |a−p|+|b−q|. If you try to place the device inside the enemy base, you will be captured by the enemy. Because of this, you need to make sure to never place the device inside the enemy base.

Unfortunately, the device is powered by a battery and you can't recharge it. This means that you can use the device at most 40 times.

Input

The input contains the answers to your queries.

Interaction

Your code is allowed to place the device on any square in the grid by writing "? i j" (1 ≤ i,j ≤ 10^9). In return, it will recieve the manhattan distance to the closest square of the enemy base from square (i,j) or -1 if the square you placed the device on is inside the enemy base or outside the grid.

If you recieve -1 instead of a positive number, exit immidiately and you will see the wrong answer verdict. Otherwise, you can get an arbitrary verdict because your solution will continue to read from a closed stream.

Your solution should use no more than 40 queries.

Once you are sure where the enemy base is located, you should print "! x y p q" (1 ≤ x ≤ p≤ 10^9, 1 ≤ y ≤ q≤ 10^9), where (x, y) is the square inside the enemy base with the smallest x and y coordinates, and (p, q) is the square inside the enemy base with the largest x and y coordinates. Note that answering doesn't count as one of the 40 queries.

After printing a query or printing the answer, do not forget to output end of line and flush the output. Otherwise, you will get idleness limit exceeded. To do this, use:

  * fflush(stdout) or cout.flush() in C++;
  * System.out.flush() in Java;
  * flush(output) in Pascal;
  * stdout.flush() in Python;
  * See the documentation for other languages.

Example

Input

1
1
2
1

Output

? 2 2
? 5 5
? 4 7
? 1 5
! 2 3 4 5


Question:
You are given an undirected graph of N nodes and M edges, E_1, E_2, ... E_M.

A connected graph is a cactus if each of it's edges belogs to at most one simple cycle. A graph is a desert if each of it's connected components is a cactus.

Find the number of pairs (L, R), (1 ≤ L ≤ R ≤ M) such that, if we delete all the edges except for E_L, E_{L+1}, ... E_R, the graph is a desert.

Input

The first line contains two integers N and M (2 ≤ N ≤ 2.5 × 10^5, 1 ≤ M ≤ 5 × 10^5). Each of the next M lines contains two integers. The i-th line describes the i-th edge. It contains integers U_i and V_i, the nodes connected by the i-th edge (E_i=(U_i, V_i)). It is guaranteed that 1 ≤ U_i, V_i ≤ N and U_i ≠ V_i.

Output

The output contains one integer number – the answer.

Examples

Input

5 6
1 2
2 3
3 4
4 5
5 1
2 4

Output

20

Input

2 3
1 2
1 2
1 2

Output

5

Note

In the second example: Graphs for pairs (1, 1), (2, 2) and (3, 3) are deserts because they don't have any cycles. Graphs for pairs (1, 2) and (2, 3) have one cycle of length 2 so they are deserts.


Question:
Bob really likes playing with arrays of numbers. That's why for his birthday, his friends bought him a really interesting machine – an array beautifier.

The array beautifier takes an array A consisting of N integers, and it outputs a new array B of length N that it constructed based on the array given to it. The array beautifier constructs the new array in the following way: it takes two numbers at different indices from the original array and writes their sum to the end of the new array. It does this step N times - resulting in an output array of length N. During this process, the machine can take the same index multiple times in different steps.

Bob was very excited about the gift that his friends gave him, so he put his favorite array in the machine. However, when the machine finished, Bob was not happy with the resulting array. He misses his favorite array very much, and hopes to get it back.

Given the array that the machine outputted, help Bob find an array that could be the original array that he put in the machine. Sometimes the machine makes mistakes, so it is possible that no appropriate input array exists for the array it has outputted. In such case, let Bob know that his array is forever lost.

Input

The first line contains one positive integer N (2 ≤ N ≤ 10^3) – the length of Bob's array.

The second line contains N integers B_1, B_2, ..., B_N (1 ≤ B_i ≤ 10^6) – the elements of the array the machine outputted.

Output

If an appropriate input array exists, print "YES", followed by the input array A_1, A_2, ..., A_N (-10^9 ≤ A_i ≤ 10^9) in the next line. Otherwise, print "NO".

Examples

Input

2
5 5

Output

YES
2 3

Input

3
1 2 3

Output

YES
0 1 2

Input

3
2 4 5

Output

NO

Input

4
1 3 5 7

Output

YES
6 -3 4 1


Question:
Alice and Bob are playing a game. They are given an array A of length N. The array consists of integers. They are building a sequence together. In the beginning, the sequence is empty. In one turn a player can remove a number from the left or right side of the array and append it to the sequence. The rule is that the sequence they are building must be strictly increasing. The winner is the player that makes the last move. Alice is playing first. Given the starting array, under the assumption that they both play optimally, who wins the game?

Input

The first line contains one integer N (1 ≤ N ≤ 2*10^5) - the length of the array A.

The second line contains N integers A_1, A_2,...,A_N (0 ≤ A_i ≤ 10^9)

Output

The first and only line of output consists of one string, the name of the winner. If Alice won, print "Alice", otherwise, print "Bob".

Examples

Input

1
5

Output

Alice

Input

3
5 4 5

Output

Alice

Input

6
5 8 2 1 10 9

Output

Bob


Question:
On the great island of Baltia, there live N people, numbered from 1 to N. There are exactly M pairs of people that are friends with each other. The people of Baltia want to organize a successful party, but they have very strict rules on what a party is and when the party is successful. On the island of Baltia, a party is a gathering of exactly 5 people. The party is considered to be successful if either all the people at the party are friends with each other (so that they can all talk to each other without having to worry about talking to someone they are not friends with) or no two people at the party are friends with each other (so that everyone can just be on their phones without anyone else bothering them). Please help the people of Baltia organize a successful party or tell them that it's impossible to do so.

Input

The first line contains two integer numbers, N (5 ≤ N ≤ 2*10^5) and M (0 ≤ M ≤ 2*10^5) – the number of people that live in Baltia, and the number of friendships. The next M lines each contains two integers U_i and V_i (1 ≤ U_i,V_i ≤ N) – meaning that person U_i is friends with person V_i. Two friends can not be in the list of friends twice (no pairs are repeated) and a person can be friends with themselves (U_i ≠ V_i).

Output

If it's possible to organize a successful party, print 5 numbers indicating which 5 people should be invited to the party. If it's not possible to organize a successful party, print -1 instead. If there are multiple successful parties possible, print any.

Examples

Input

6 3
1 4
4 2
5 4

Output

1 2 3 5 6

Input

5 4
1 2
2 3
3 4
4 5

Output

-1


Question:
You have received data from a Bubble bot. You know your task is to make factory facilities, but before you even start, you need to know how big the factory is and how many rooms it has. When you look at the data you see that you have the dimensions of the construction, which is in rectangle shape: N x M.

Then in the next N lines you have M numbers. These numbers represent factory tiles and they can go from 0 to 15. Each of these numbers should be looked in its binary form. Because from each number you know on which side the tile has walls. For example number 10 in it's binary form is 1010, which means that it has a wall from the North side, it doesn't have a wall from the East, it has a wall on the South side and it doesn't have a wall on the West side. So it goes North, East, South, West.

It is guaranteed that the construction always has walls on it's edges. The input will be correct.

Your task is to print the size of the rooms from biggest to smallest.

Input

The first line has two numbers which are N and M, the size of the construction. Both are integers:

n (1 ≤ n ≤ 10^3)

m (1 ≤ m ≤ 10^3)

Next N x M numbers represent each tile of construction.

Output

Once you finish processing the data your output consists of one line sorted from biggest to smallest room sizes.

Example

Input

4 5
9 14 11 12 13
5 15 11 6 7
5 9 14 9 14
3 2 14 3 14

Output

9 4 4 2 1


Question:
You are given array a_1, a_2, …, a_n, consisting of non-negative integers.

Let's define operation of "elimination" with integer parameter k (1 ≤ k ≤ n) as follows:

  * Choose k distinct array indices 1 ≤ i_1 < i_2 < … < i_k ≤ n.
  * Calculate x = a_{i_1} ~ \& ~ a_{i_2} ~ \& ~ … ~ \& ~ a_{i_k}, where \& denotes the [bitwise AND operation](https://en.wikipedia.org/wiki/Bitwise_operation#AND) (notes section contains formal definition).
  * Subtract x from each of a_{i_1}, a_{i_2}, …, a_{i_k}; all other elements remain untouched.

Find all possible values of k, such that it's possible to make all elements of array a equal to 0 using a finite number of elimination operations with parameter k. It can be proven that exists at least one possible k for any array a.

Note that you firstly choose k and only after that perform elimination operations with value k you've chosen initially.

Input

Each test contains multiple test cases. The first line contains the number of test cases t (1 ≤ t ≤ 10^4). Description of the test cases follows.

The first line of each test case contains one integer n (1 ≤ n ≤ 200 000) — the length of array a.

The second line of each test case contains n integers a_1, a_2, …, a_n (0 ≤ a_i < 2^{30}) — array a itself.

It's guaranteed that the sum of n over all test cases doesn't exceed 200 000.

Output

For each test case, print all values k, such that it's possible to make all elements of a equal to 0 in a finite number of elimination operations with the given parameter k.

Print them in increasing order.

Example

Input

5
4
4 4 4 4
4
13 7 25 19
6
3 5 3 1 7 1
1
1
5
0 0 0 0 0

Output

1 2 4
1 2
1
1
1 2 3 4 5

Note

In the first test case:

  * If k = 1, we can make four elimination operations with sets of indices \{1\}, \{2\}, \{3\}, \{4\}. Since \& of one element is equal to the element itself, then for each operation x = a_i, so a_i - x = a_i - a_i = 0.
  * If k = 2, we can make two elimination operations with, for example, sets of indices \{1, 3\} and \{2, 4\}: x = a_1 ~ \& ~ a_3 = a_2 ~ \& ~ a_4 = 4 ~ \& ~ 4 = 4. For both operations x = 4, so after the first operation a_1 - x = 0 and a_3 - x = 0, and after the second operation — a_2 - x = 0 and a_4 - x = 0.
  * If k = 3, it's impossible to make all a_i equal to 0. After performing the first operation, we'll get three elements equal to 0 and one equal to 4. After that, all elimination operations won't change anything, since at least one chosen element will always be equal to 0.
  * If k = 4, we can make one operation with set \{1, 2, 3, 4\}, because x = a_1 ~ \& ~ a_2 ~ \& ~ a_3 ~ \& ~ a_4 = 4.

In the second test case, if k = 2 then we can make the following elimination operations:

  * Operation with indices \{1, 3\}: x = a_1 ~ \& ~ a_3 = 13 ~ \& ~ 25 = 9. a_1 - x = 13 - 9 = 4 and a_3 - x = 25 - 9 = 16. Array a will become equal to [4, 7, 16, 19].
  * Operation with indices \{3, 4\}: x = a_3 ~ \& ~ a_4 = 16 ~ \& ~ 19 = 16. a_3 - x = 16 - 16 = 0 and a_4 - x = 19 - 16 = 3. Array a will become equal to [4, 7, 0, 3].
  * Operation with indices \{2, 4\}: x = a_2 ~ \& ~ a_4 = 7 ~ \& ~ 3 = 3. a_2 - x = 7 - 3 = 4 and a_4 - x = 3 - 3 = 0. Array a will become equal to [4, 4, 0, 0].
  * Operation with indices \{1, 2\}: x = a_1 ~ \& ~ a_2 = 4 ~ \& ~ 4 = 4. a_1 - x = 4 - 4 = 0 and a_2 - x = 4 - 4 = 0. Array a will become equal to [0, 0, 0, 0].

Formal definition of bitwise AND:

Let's define bitwise AND (\&) as follows. Suppose we have two non-negative integers x and y, let's look at their binary representations (possibly, with leading zeroes): x_k ... x_2 x_1 x_0 and y_k ... y_2 y_1 y_0. Here, x_i is the i-th bit of number x, and y_i is the i-th bit of number y. Let r = x ~ \& ~ y is a result of operation \& on number x and y. Then binary representation of r will be r_k ... r_2 r_1 r_0, where:

$$$ r_i = \begin{cases} 1, ~ if ~ x_i = 1 ~ and ~ y_i = 1 \\\ 0, ~ if ~ x_i = 0 ~ or ~ y_i = 0 \end{cases} $$$


Question:
Frog Gorf is traveling through Swamp kingdom. Unfortunately, after a poor jump, he fell into a well of n meters depth. Now Gorf is on the bottom of the well and has a long way up.

The surface of the well's walls vary in quality: somewhere they are slippery, but somewhere have convenient ledges. In other words, if Gorf is on x meters below ground level, then in one jump he can go up on any integer distance from 0 to a_x meters inclusive. (Note that Gorf can't jump down, only up).

Unfortunately, Gorf has to take a break after each jump (including jump on 0 meters). And after jumping up to position x meters below ground level, he'll slip exactly b_x meters down while resting.

Calculate the minimum number of jumps Gorf needs to reach ground level.

Input

The first line contains a single integer n (1 ≤ n ≤ 300 000) — the depth of the well.

The second line contains n integers a_1, a_2, …, a_n (0 ≤ a_i ≤ i), where a_i is the maximum height Gorf can jump from i meters below ground level.

The third line contains n integers b_1, b_2, …, b_n (0 ≤ b_i ≤ n - i), where b_i is the distance Gorf will slip down if he takes a break on i meters below ground level.

Output

If Gorf can't reach ground level, print -1. Otherwise, firstly print integer k — the minimum possible number of jumps.

Then print the sequence d_1,\,d_2, …,\,d_k where d_j is the depth Gorf'll reach after the j-th jump, but before he'll slip down during the break. Ground level is equal to 0.

If there are multiple answers, print any of them.

Examples

Input

3
0 2 2
1 1 0

Output

2
1 0

Input

2
1 1
1 0

Output

-1

Input

10
0 1 2 3 5 5 6 7 8 5
9 8 7 1 5 4 3 2 0 0

Output

3
9 4 0

Note

In the first example, Gorf is on the bottom of the well and jump to the height 1 meter below ground level. After that he slip down by meter and stays on height 2 meters below ground level. Now, from here, he can reach ground level in one jump.

In the second example, Gorf can jump to one meter below ground level, but will slip down back to the bottom of the well. That's why he can't reach ground level.

In the third example, Gorf can reach ground level only from the height 5 meters below the ground level. And Gorf can reach this height using a series of jumps 10 ⇒ 9 \dashrightarrow 9 ⇒ 4 \dashrightarrow 5 where ⇒ is the jump and \dashrightarrow is slipping during breaks.