# Yet another MEX problem solution codechef

The MEX (minimum excluded) of an array is the smallest non-negative integer that does not belong to the array. For instance:

• The MEX of [2,2,1][2,2,1] is 00, because 00 does not belong to the array.
• The MEX of [3,1,0,1][3,1,0,1] is 22, because 00 and 11 belong to the array, but 22 does not.
• The MEX of [0,3,1,2][0,3,1,2] is 44, because 0,1,20,1,2 and 33 belong to the array, but 44 does not.

## Yet another MEX problem solution codechef

You are given an array AA of length NN. You create a list BB consisting of the MEX-es of all subarrays of the array AA. Formally, for all pairs (l,r)(l,r) such that 1lrN1≤l≤r≤N, you calculate MEX(Al,Al+1,,Ar)MEX(Al,Al+1,…,Ar) and append the value in the list BB. Find the KK-th smallest value in the list BB.

Note: Since the size of the input and output is large, please use fast input-output methods.

### Input Format Yet another MEX problem solution codechef

• The first line contains TT denoting the number of test cases. Then the test cases follow.
• The first line of each test case contains two space-separated integers NN and KK.
• The second line contains NN space-separated integers A1,A2,,ANA1,A2,…,AN denoting the given array.

### Output Format Yet another MEX problem solution codechef

For each test case, output on a single line the KK-th smallest value in the list BB.

### Constraints Yet another MEX problem solution codechef

• 1T31041≤T≤3⋅104
• 1N1051≤N≤105
• 1KN(N+1)2 1≤K≤N⋅(N+1)2
• 0AiN0≤Ai≤N
• Sum of NN over all test cases does not exceed 21062⋅106.

### Subtasks Yet another MEX problem solution codechef

Subtask 1 (10 points): Yet another MEX problem solution codechef

• 1N51031≤N≤5⋅103
• Sum of NN over all test cases does not exceed 51045⋅104.

Subtask 2 (90 points): Original constraints

### Sample Input 1  Yet another MEX problem solution codechef

3
3 4
1 0 2
3 2
2 1 3
3 6
0 1 2


### Sample Output 1 Yet another MEX problem solution codechef

1
0
3


## Explanation Yet another MEX problem solution codechef

Test case 11: MEX(A1)=0MEX(A1)=0MEX(A1,A2)=2MEX(A1,A2)=2MEX(A1,A2,A3)=3MEX(A1,A2,A3)=3MEX(A2)=1MEX(A2)=1MEX(A2,A3)=1MEX(A2,A3)=1MEX(A3)=0MEX(A3)=0. Hence the list B=[0,2,3,1,1,0]B=[0,2,3,1,1,0] and the 44-th smallest value in BB is 11.

Test case 22: The MEX of all subarrays of the array AA is 00. Hence the 22-nd smallest element in the list BB is 00.

The MEX (minimum excluded) of an array is the smallest non-negative integer that does not belong to the array. For instance:

• The MEX of [2,2,1][2,2,1] is 00, because 00 does not belong to the array.
• The MEX of [3,1,0,1][3,1,0,1] is 22, because 00 and 11 belong to the array, but 22 does not.
• The MEX of [0,3,1,2][0,3,1,2] is 44, because 0,1,20,1,2 and 33 belong to the array, but 44 does not.

## Yet another MEX problem solution codechef

You are given an array AA of length NN. You create a list BB consisting of the MEX-es of all subarrays of the array AA. Formally, for all pairs (l,r)(l,r) such that 1lrN1≤l≤r≤N, you calculate MEX(Al,Al+1,,Ar)MEX(Al,Al+1,…,Ar) and append the value in the list BB. Find the KK-th smallest value in the list BB.

Note: Since the size of the input and output is large, please use fast input-output methods.

### Input Format Yet another MEX problem solution codechef

• The first line contains TT denoting the number of test cases. Then the test cases follow.
• The first line of each test case contains two space-separated integers NN and KK.
• The second line contains NN space-separated integers A1,A2,,ANA1,A2,…,AN denoting the given array.

### Output Format Yet another MEX problem solution codechef

For each test case, output on a single line the KK-th smallest value in the list BB.

### Constraints Yet another MEX problem solution codechef

• 1T31041≤T≤3⋅104
• 1N1051≤N≤105
• 1KN(N+1)2 1≤K≤N⋅(N+1)2
• 0AiN0≤Ai≤N
• Sum of NN over all test cases does not exceed 21062⋅106.

### Subtasks Yet another MEX problem solution codechef

Subtask 1 (10 points): Yet another MEX problem solution codechef

• 1N51031≤N≤5⋅103
• Sum of NN over all test cases does not exceed 51045⋅104.

Subtask 2 (90 points): Original constraints

### Sample Input 1  Yet another MEX problem solution codechef

3
3 4
1 0 2
3 2
2 1 3
3 6
0 1 2


### Sample Output 1 Yet another MEX problem solution codechef

1
0
3


## Explanation Yet another MEX problem solution codechef

Test case 11: MEX(A1)=0MEX(A1)=0MEX(A1,A2)=2MEX(A1,A2)=2MEX(A1,A2,A3)=3MEX(A1,A2,A3)=3MEX(A2)=1MEX(A2)=1MEX(A2,A3)=1MEX(A2,A3)=1MEX(A3)=0MEX(A3)=0. Hence the list B=[0,2,3,1,1,0]B=[0,2,3,1,1,0] and the 44-th smallest value in BB is 11.

Test case 22: The MEX of all subarrays of the array AA is 00. Hence the 22-nd smallest element in the list BB is 00.

Finally, progress reached the Madoka family and she decided to play with her little sister in the sensational game Space Arrays.

## Yet another MEX problem solution codechef

The rules of the game are as follows:

• Initially, a sequence a1,a2,,aNa1,a2,…,aN is given.
• The players alternate turns.
• In each turn, the current player must choose an index ii and increment aiai, i.e. change aiai to ai+1ai+1.
• Afterwards, if there is no permutation p1,p2,,pNp1,p2,…,pN of the integers 11 through NN such that aipiai≤pi holds for each valid ii, the current player loses.
• Otherwise, the game continues with the next turn.

Madoka is asking you to help her ― tell her if the first player (the player that plays in the first turn) or second player wins this game if both play optimally.

### Input Yet another MEX problem solution codechef

• The first line of the input contains a single integer TT denoting the number of test cases. The description of TT test cases follows.
• The first line of each test case contains a single integer NN.
• The second line contains NN space-separated integers a1,a2,,aNa1,a2,…,aN.

### Output Yet another MEX problem solution codechef

For each test case, print a single line containing the string "First" if the first player wins or "Second" if the second player wins (without quotes).

### Constraints Yet another MEX problem solution codechef

• 1T21041≤T≤2⋅104
• 1N21051≤N≤2⋅105
• The sum of NN over all test cases doesn’t exceed 21052⋅105
• 1aiN1≤ai≤N for each valid ii

### Subtasks Yet another MEX problem solution codechef

Subtask #1 (100 points): Original constraints

### Sample Input 1  Yet another MEX problem solution codechef

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


### Sample Output 1  Yet another MEX problem solution codechef

First
Second
Second
Second


### Explanation Yet another MEX problem solution codechef

Example case 1:

• If the first player increases the fourth element, the resulting sequence is (1,2,3,4)(1,2,3,4). The second player loses after increasing any of the elements.Yet another MEX problem solution codechef
• If the first player increases the second element, the resulting sequence is (1,3,3,3)(1,3,3,3), and he loses because there is no valid permutation. For example if p=(2,1,4,3)p=(2,1,4,3), the second element of aa is greater than the second element of pp.