Ntarsis' Set

Description:

Input:

Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 10^5$$$). The description of the test cases follows.

The first line of each test case consists of two integers $$$n$$$ and $$$k$$$ ($$$1 \leq n,k \leq 2 \cdot 10^5$$$) — the length of $$$a$$$ and the number of days.

The following line of each test case consists of $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ ($$$1 \leq a_i \leq 10^9$$$) — the elements of array $$$a$$$.

It is guaranteed that:

• The sum of $$$n$$$ over all test cases won't exceed $$$2 \cdot 10^5$$$;
• The sum of $$$k$$$ over all test cases won't exceed $$$2 \cdot 10^5$$$;
• $$$a_1 < a_2 < \cdots < a_n$$$ for all test cases.

Output:

For each test case, print an integer that is the smallest element in $$$S$$$ after $$$k$$$ days.

Sample Input:

7
5 1
1 2 4 5 6
5 3
1 3 5 6 7
4 1000
2 3 4 5
9 1434
1 4 7 9 12 15 17 18 20
10 4
1 3 5 7 9 11 13 15 17 19
10 6
1 4 7 10 13 16 19 22 25 28
10 150000
1 3 4 5 10 11 12 13 14 15

Sample Output:

3
9
1
12874
16
18
1499986

Note:

For the first test case, each day the $$$1$$$-st, $$$2$$$-nd, $$$4$$$-th, $$$5$$$-th, and $$$6$$$-th smallest elements need to be removed from $$$S$$$. So after the first day, $$$S$$$ will become $$$\require{cancel}$$$ $$$\{\cancel 1, \cancel 2, 3, \cancel 4, \cancel 5, \cancel 6, 7, 8, 9, \ldots\} = \{3, 7, 8, 9, \ldots\}$$$. The smallest element is $$$3$$$.

For the second case, each day the $$$1$$$-st, $$$3$$$-rd, $$$5$$$-th, $$$6$$$-th and $$$7$$$-th smallest elements need to be removed from $$$S$$$. $$$S$$$ will be changed as follows:

Day	$$$S$$$ before		$$$S$$$ after
1	$$$\{\cancel 1, 2, \cancel 3, 4, \cancel 5, \cancel 6, \cancel 7, 8, 9, 10, \ldots \}$$$	$$$\to$$$	$$$\{2, 4, 8, 9, 10, \ldots\}$$$
2	$$$\{\cancel 2, 4, \cancel 8, 9, \cancel{10}, \cancel{11}, \cancel{12}, 13, 14, 15, \ldots\}$$$	$$$\to$$$	$$$\{4, 9, 13, 14, 15, \ldots\}$$$
3	$$$\{\cancel 4, 9, \cancel{13}, 14, \cancel{15}, \cancel{16}, \cancel{17}, 18, 19, 20, \ldots\}$$$	$$$\to$$$	$$$\{9, 14, 18, 19, 20, \ldots\}$$$

The smallest element left after $$$k = 3$$$ days is $$$9$$$.