F. Lisa and the Martians
time limit per test
3 seconds
memory limit per test
512 megabytes
input
standard input
output
standard output

Lisa was kidnapped by martians! It okay, because she has watched a lot of TV shows about aliens, so she knows what awaits her. Let's call integer martian if it is a non-negative integer and strictly less than $2^k$, for example, when $k = 12$, the numbers $51$, $1960$, $0$ are martian, and the numbers $\pi$, $-1$, $\frac{21}{8}$, $4096$ are not.

The aliens will give Lisa $n$ martian numbers $a_1, a_2, \ldots, a_n$. Then they will ask her to name any martian number $x$. After that, Lisa will select a pair of numbers $a_i, a_j$ ($i \neq j$) in the given sequence and count $(a_i \oplus x) \& (a_j \oplus x)$. The operation $\oplus$ means Bitwise exclusive OR, the operation $\&$ means Bitwise And. For example, $(5 \oplus 17) \& (23 \oplus 17) = (00101_2 \oplus 10001_2) \& (10111_2 \oplus 10001_2) = 10100_2 \& 00110_2 = 00100_2 = 4$.

Lisa is sure that the higher the calculated value, the higher her chances of returning home. Help the girl choose such $i, j, x$ that maximize the calculated value.

Input

The first line contains an integer $t$ ($1 \le t \le 10^4$) — number of testcases.

Each testcase is described by two lines.

The first line contains integers $n, k$ ($2 \le n \le 2 \cdot 10^5$, $1 \le k \le 30$) — the length of the sequence of martian numbers and the value of $k$.

The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($0 \le a_i < 2^k$) — a sequence of martian numbers.

It is guaranteed that the sum of $n$ over all testcases does not exceed $2 \cdot 10^5$.

Output

For each testcase, print three integers $i, j, x$ ($1 \le i, j \le n$, $i \neq j$, $0 \le x < 2^k$). The value of $(a_i \oplus x) \& (a_j \oplus x)$ should be the maximum possible.

If there are several solutions, you can print any one.

Example
Input
10
5 4
3 9 1 4 13
3 1
1 0 1
6 12
144 1580 1024 100 9 13
4 3
7 3 0 4
3 2
0 0 1
2 4
12 2
9 4
6 14 9 4 4 4 5 10 2
2 1
1 0
2 4
11 4
9 4
2 11 10 1 6 9 11 0 5
Output
1 3 14
1 3 0
5 6 4082
2 3 7
1 2 3
1 2 15
4 5 11
1 2 0
1 2 0
2 7 4
Note

First testcase: $(3 \oplus 14) \& (1 \oplus 14) = (0011_2 \oplus 1110_2) \& (0001_2 \oplus 1110_2) = 1101_2 = 1101_2 \& 1111_2 = 1101_2 = 13$.

Second testcase: $(1 \oplus 0) \& (1 \oplus 0) = 1$.

Third testcase: $(9 \oplus 4082) \& (13 \oplus 4082) = 4091$.

Fourth testcase: $(3 \oplus 7) \& (0 \oplus 7) = 4$.