作者： chenchen

Farmer John's cows like nothing more than cereal for breakfast! In fact, the cows have such large appetites that they will each eat an entire box of cereal for a single meal.
The farm has recently received a shipment with M different types of cereal (2≤M≤10^5). Unfortunately, there is only one box of each cereal! Each of the N cows (1≤N≤10^5) has a favorite cereal and a second favorite cereal. When given a selection of cereals to choose from, a cow performs the following process:
If the box of her favorite cereal is still available, take it and leave.
Otherwise, if the box of her second-favorite cereal is still available, take it and leave.
Otherwise, she will moo with disappointment and leave without taking any cereal.
Find the minimum number of cows that go hungry if you permute them optimally. Also, find any permutation of the N cows that achieves this minimum.
INPUT FORMAT (input arrives from the terminal / stdin):
The first line contains two space-separated integers N and M.
For each 1≤i≤N, the i-th line contains two space-separated integers fi and si (1≤fi,si≤M and fi≠si) denoting the favorite and second-favorite cereals of the i-th cow.
OUTPUT FORMAT (print output to the terminal / stdout):
Print the minimum number of cows that go hungry, followed by any permutation of 1…N that achieves this minimum. If there are multiple permutations, any one will be accepted.
SAMPLE INPUT:
8 10
2 1
3 4
2 3
6 5
7 8
6 7
7 5
5 8
SAMPLE OUTPUT:
1
1
3
2
8
4
6
5
7
In this example, there are 8 cows and 10 types of cereal.
Note that we can effectively solve for the first three cows independently of the last five, since they share no favorite cereals in common.
If the first three cows choose in the order [1,2,3], then cow 1 will choose cereal 2, cow 2 will choose cereal 3, and cow 3 will go hungry.
If the first three cows choose in the order [1,3,2], then cow 1 will choose cereal 2, cow 3 will choose cereal 3, and cow 2 will choose cereal 4; none of these cows will go hungry.
Of course, there are other permutations that result in none of the first three cows going hungry. For example, if the first three cows choose in the order [3,1,2] then cow 3 will choose cereal 2, cow 1 will choose cereal 1, and cow 2 will choose cereal 3; again, none of cows [1,2,3] will go hungry.
It can be shown that out of the last five cows, at least one must go hungry.
SCORING:
In 4 out of 14 test cases, N,M≤10^0.
In 10 out of 14 test cases, no additional constraints.

Farmer John's N cows (N≤3×10^5) have heights 1,2,…,N. One day, the cows are standing in a line in some order playing frisbee; let h1…hN denote the heights of the cows in this order (so the h's are a permutation of 1…N).
Two cows at positions i and j in the line can successfully throw the frisbee back and forth if and only if every cow between them has height lower than min(hi,hj).
Please compute the sum of distances between all pairs of locations i<j at which there resides a pair of cows that can successfully throw the frisbee back and forth. The distance between locations i and j is j?i+1.
INPUT FORMAT (input arrives from the terminal / stdin):
The first line of input contains a single integer N. The next line of input contains h1…hN, separated by spaces.
OUTPUT FORMAT (print output to the terminal / stdout):
Output the sum of distances of all pairs of locations at which there are cows that can throw the frisbee back and forth. Note that the large size of integers involved in this problem may require the use of 64-bit integer data types (e.g., a "long long" in C/C++).
SAMPLE INPUT:
7
4 3 1 2 5 6 7
SAMPLE OUTPUT:
24
The pairs of successful locations in this example are as follows:
(1, 2), (1, 5), (2, 3), (2, 4), (2, 5), (3, 4), (4, 5), (5, 6), (6, 7)
SCORING
Test cases 1-3 satisfy N≤5000.
Test cases 4-11 satisfy no additional constraints.

Farmer John's cows each want to find their soulmate -- another cow with similar characteristics with whom they are maximally compatible. Each cow's personality is described by an integer pi (1≤pi≤10^18). Two cows with the same personality are soulmates. A cow can change her personality via a "change operation" by multiplying by 2, dividing by 2 (if pi is even), or adding 1.
Farmer John initially pairs his cows up in an arbitrary way. He is curious how many change operations would be needed to make each pair of cows into soulmates. For each pairing, decide the minimum number of change operations the first cow in the pair must make to become soulmates with the second cow.
INPUT FORMAT (input arrives from the terminal / stdin):
The first line contains N (1≤N≤10), the number of pairs of cows. Each of the remaining N lines describes a pair of cows in terms of two integers giving their personalities. The first number indicates the personality of the cow that must be changed to match the second.
OUTPUT FORMAT (print output to the terminal / stdout):
Please write N lines of output. For each pair, print the minimum number of operations required for the first cow to make her personality match that of the second.
SAMPLE INPUT:
6
31 13
12 8
25 6
10 24
1 1
997 120
SAMPLE OUTPUT:
8
3
8
3
0
20
For the first test case, an optimal sequence of changes is 31⟹32⟹16⟹8⟹9⟹10⟹11⟹12⟹13.
For the second test case, an optimal sequence of changes is 12⟹6⟹7⟹8.
SCORING:
Test cases 1-4 satisfy pi≤10^5.
Test cases 5-12 satisfy no additional constraints.

Farmer John's cows have decided to offer a programming contest for the cows on Farmer Nhoj's farm. In order to make the problems as fun as possible, they have spent considerable time coming up with challenging input cases. For one problem in particular, "Haybales", the cows need your help devising challenging inputs. This involve solving the following somewhat intriguing problem:
There is an array of sorted integers x1≤x2≤?≤xN (1≤N≤10^5), and an integer K. You don't know the array or K, but you do know for each index i, the largest index ji such that xji≤xi+K. It is guaranteed that i≤ji and j1≤j2≤?≤jN≤N.
Given this information, Farmer John's cows need to construct any array along with some integer K that matches that information. The construction needs to satisfy 0≤xi≤10^18 for all i and 1≤K≤10^18.
It can be proven that this is always possible. Help Farmer John's cows solve this problem!
INPUT FORMAT (input arrives from the terminal / stdin):
The first line of input contains N. The next line contains j1,j2,…,jN.
OUTPUT FORMAT (print output to the terminal / stdout):
Print K, then x1,…,xN on separate lines. Any valid output will be accepted.
SAMPLE INPUT:
6
2 2 4 5 6 6
SAMPLE OUTPUT:
6
1
6
17
22
27
32
The sample output is the array a=[1,6,17,22,27,32] with K=6. j1=2 is satisfied because a2=6≤1+6=a1+K but a3=17>1+6=a1+K, so a2 is the largest element that is at most a1. Similarly,
j2=2 is satisfied because a2=6≤6+6 but a3=17>6+6
j3=4 is satisfied because a4=22≤17+6 but a5=27>17+6
j4=5 is satisfied because a5=27≤22+6 but a5=32>22+6
j5=6 is satisfied because a6=32≤27+6 and a6 is the last element of the array
j6=6 is satisfied because a6=32≤32+6 and a6 is the last element of the array
This is not the only possible correct output for the sample input. For example, you could instead output the array [1,2,4,5,6,7] with K=1.
SCORING:
For 50% of all inputs, N≤5000
For the remaining inputs, there are no additional constraints.

Farmer John operates a collection of N farms (1≤N≤2·10^5), conveniently numbered 1…N. Initially, there are no roads connecting these farms to each-other, and each farm is actively producing milk.
Due to the dynamic nature of the economy, Farmer John needs to make changes to his farms according to a series of Q update operations (0≤Q≤2⋅10^5). Update operations come in three possible forms:
(D x) Deactivate an active farm x, so it no longer produces milk.
(A x y) Add a road between two active farms x and y.
(R e) Remove the eth road that was previously added (e=1 is the first road that was added).
A farm x that is actively producing milk, or that can reach another active farm via a series of roads, is called a "relevant" farm. For each farm x, please calculate the maximum i (0≤i≤Q) such that x is relevant after the i-th update.
INPUT FORMAT (input arrives from the terminal / stdin):
The first line of input contains N and Q. The next Q lines each contain an update of one of the following forms:
D x
A x y
R e
It is guaranteed that for updates of type R, e is at most the number of roads that have been added so far, and no two updates of type R have the same value of e.
OUTPUT FORMAT (print output to the terminal / stdout):
Please output N lines, each containing an integer in the range 0…Q.
SAMPLE INPUT:
5 9
A 1 2
A 2 3
D 1
D 3
A 2 4
D 2
R 2
R 1
R 3
SAMPLE OUTPUT:
7
8
6
9
9
In this example, roads are removed in the order (2,3),(1,2),(2,4).
Farm 1 is relevant just before (1,2) is removed.
Farm 2 is relevant just before (2,4) is removed.
Farm 3 is relevant just before (2,3) is removed.
Farms 4 and 5 are still active after all queries. Therefore they both stay relevant, and the output for both should be Q.
SCORING:
Tests 2 through 5 satisfy N≤103, Q≤2⋅103
Test cases 6 through 20 satisfy no additional constraints.

The grass has dried up in Farmer John's pasture due to a drought. After hours of despair and contemplation, FJ comes up with the brilliant idea of purchasing corn to feed his precious cows.
FJ’s N (1≤N≤100) cows are arranged in a line such that the ith cow in line has a non-negative integer hunger level of hi. As FJ’s cows are social animals and insist on eating together, the only way FJ can decrease the hunger levels of his cows is to select two adjacent cows i and i+1 and feed each of them a bag of corn, causing each of their hunger levels to decrease by one.
FJ wants to feed his cows until all of them have the same non-negative hunger level. Although he doesn't know his cows' exact hunger levels, he does know an upper bound on the hunger level of each cow; specifically, the hunger level hi of the i-th cow is at most Hi (0≤Hi≤1000).
Your job is to count the number of N-tuples of hunger levels [h1,h2,…,hN] that are consistent with these upper bounds such that it is possible for FJ to achieve his goal, modulo 10^9+7.
INPUT FORMAT (input arrives from the terminal / stdin):
The first line contains N.
The second line contains H1,H2,…,HN.
OUTPUT FORMAT (print output to the terminal / stdout):
The number of N-tuples of hunger levels modulo 10^9+7.
SAMPLE INPUT:
3
9 11 7
SAMPLE OUTPUT:
241
There are (9+1)⋅ (11+1)⋅ (7+1) 3-tuples h that are consistent with H.
One of these tuples is h=[8,10,5]. In this case, it is possible to make all cows have equal hunger values: give two bags of corn to both cows 2 and 3, then give five bags of corn to both cows 1 and 2, resulting in each cow having a hunger level of 3.
Another one of these tuples is h=[0,1,0]. In this case, it is impossible to make the hunger levels of the cows equal.
SAMPLE INPUT:
4
6 8 5 9
SAMPLE OUTPUT:
137
SCORING:
N is even in even-numbered tests and odd in odd-numbered tests.
Tests 3 and 4 satisfy N≤6 and Hi≤10.
Tests 5 through 10 satisfy N≤50 and Hi≤100.
Tests 11 through 20 satisfy no further constraints.

The cows are taking a multiple choice test. But instead of a standard test where your selected choices are scored for each question individually and then summed, in this test your selected choices are summed before being scored.
Specifically, you are given N (2≤N≤10*5) groups of integer vectors on the 2D plane, where each vector is denoted by an ordered pair (x,y). Choose one vector from each group such that the sum of the vectors is as far away from the origin as possible.
It is guaranteed that the total number of vectors is at most 2?105. Each group has size at least 2, and within a group, all vectors are distinct. It is also guaranteed that every x and y coordinate has absolute value at most 109N.
INPUT FORMAT (input arrives from the terminal / stdin):
The first line contains N, the number of groups.
Each group starts with G, the number of vectors in the group, followed by G lines containing the vectors in that group. Consecutive groups are separated by newlines.
OUTPUT FORMAT (print output to the terminal / stdout):
The maximum possible squared Euclidean distance.
SAMPLE INPUT:
3
2
-2 0
1 0
2
0 -2
0 1
3
-5 -5
5 1
10 10
SAMPLE OUTPUT:
242
It is optimal to select (1,0) from the first group, (0,1) from the second group, and (10,10) from the third group. The sum of these vectors is (11,11), which is squared distance 112+112=242 from the origin.
SCORING:
In test cases 1-5, the total number of vectors is at most 103.
In test cases 6-9, every group has size exactly two.
Test cases 10-17 satisfy no additional constraints.

As usual, Bessie the cow is causing trouble in Farmer John's barn. FJ has N (1≤N≤5000) stacks of haybales. For each i∈[1,N], the ith stack has hi (1≤hi≤10^9) haybales. Bessie does not want any haybales to fall, so the only operation she can perform is as follows:
If two adjacent stacks' heights differ by exactly one, she can move the top haybale of the taller stack to the shorter stack.
How many configurations are obtainable after performing the above operation finitely many times, modulo 109+7? Two configurations are considered the same if, for all i, the ith stack has the same number of haybales in both.

INPUT FORMAT (input arrives from the terminal / stdin):
The first line contains T (1≤T≤10), the number of independent test cases, all of which must be solved to solve one input correctly.
Each test case consists of N, and then a sequence of N heights. It is guaranteed that the sum of N over all test cases does not exceed 5000.

OUTPUT FORMAT (print output to the terminal / stdout):
Please output T lines, one for each test case.
SAMPLE INPUT:
7
4
2 2 2 3
4
3 3 1 2
4
5 3 4 2
6
3 3 1 1 2 2
6
1 3 3 4 1 2
6
4 1 2 3 5 4
10
1 5 6 6 6 4 2 3 2 5
SAMPLE OUTPUT:
4
4
5
15
9
8
19
For the first test case, the four possible configurations are:

(2,2,2,3),(2,2,3,2),(2,3,2,2),(3,2,2,2).
For the second test case, the four possible configurations are:

(2,3,3,1),(3,2,3,1),(3,3,2,1),(3,3,1,2).
SCORING:
Inputs 1-3 satisfy N≤10.
Input 4 satisfies 1≤hi≤3 for all i.
Inputs 5-7 satisfy |hi−i|≤1 for all i.
Inputs 8-10 satisfy 1≤hi≤4 for all i and N≤10^0.
Inputs 11-13 satisfy N≤100.
Inputs 14-17 satisfy N≤1000.
Inputs 18-21 satisfy no additional constraints.

As usual, Bessie the cow is causing trouble in Farmer John's barn. FJ has N (1≤N≤5000) stacks of haybales. For each i∈[1,N], the ith stack has hi (1≤hi≤109) haybales. Bessie does not want any haybales to fall, so the only operation she can perform is as follows:
If two adjacent stacks' heights differ by exactly one, she can move the top haybale of the taller stack to the shorter stack.
How many configurations are obtainable after performing the above operation finitely many times, modulo 10^9+7? Two configurations are considered the same if, for all i, the ith stack has the same number of haybales in both.

INPUT FORMAT (input arrives from the terminal / stdin):
The first line contains T (1≤T≤10), the number of independent test cases, all of which must be solved to solve one input correctly.
Each test case consists of N, and then a sequence of N heights. It is guaranteed that the sum of N over all test cases does not exceed 5000.

OUTPUT FORMAT (print output to the terminal / stdout):
Please output T lines, one for each test case.
SAMPLE INPUT:
7
4
2 2 2 3
4
3 3 1 2
4
5 3 4 2
6
3 3 1 1 2 2
6
1 3 3 4 1 2
6
4 1 2 3 5 4
10
1 5 6 6 6 4 2 3 2 5
SAMPLE OUTPUT:
4
4
5
15
9
8
19
For the first test case, the four possible configurations are:

(2,2,2,3),(2,2,3,2),(2,3,2,2),(3,2,2,2).
For the second test case, the four possible configurations are:

(2,3,3,1),(3,2,3,1),(3,3,2,1),(3,3,1,2).
SCORING:
Inputs 1-3 satisfy N≤10.
Input 4 satisfies 1≤hi≤3 for all i.
Inputs 5-7 satisfy |hi−i|≤1 for all i.
Inputs 8-10 satisfy 1≤hi≤4 for all i and N≤100.
Inputs 11-13 satisfy N≤100.
Inputs 14-17 satisfy N≤1000.
Inputs 18-21 satisfy no additional constraints.
Problem credits: Daniel Zhang

Bessie is bored and yet again causing trouble in Farmer John's barn. FJ has N (1≤N≤10^5) stacks of haybales. For each i∈[1,N], the ith stack has hi (1≤hi≤10^9) haybales. Bessie does not want any haybales to fall, so the only operation she can perform is as follows:
If two adjacent stacks' heights differ by at most K (1≤K≤10^9), she can swap the two stacks.
What is the lexicographically minimum sequence of heights that Bessie can obtain after some sequence of these operations?

**Note: the time and memory limits for this problem are 4s and 512MB, twice the defaults.**

INPUT FORMAT (input arrives from the terminal / stdin):
The first line of input contains N and K. The i+1-st line contains the height of the i-th haybale.
OUTPUT FORMAT (print output to the terminal / stdout):
Please print out N lines, the i-th containing the height of the i-th haybale in the solution.
SAMPLE INPUT:
5 3
7
7
3
6
2
SAMPLE OUTPUT:
6
7
7
2
3
One way that Bessie can swap the stacks is as follows:

7 7 3 6 2
-> 7 7 6 3 2
-> 7 7 6 2 3
-> 7 6 7 2 3
-> 6 7 7 2 3
SCORING:
In 10% of all input cases, N≤10^0
In another 20% of all input cases, N≤5000
In the remaining 70% of input cases, there are no additional constraints
Problem credits: Daniel Zhang and Benjamin Qi