C. XOR-distance
time limit per test
2 seconds
memory limit per test
256 megabytes
input
standard input
output
standard output

You are given integers $$$a$$$, $$$b$$$, $$$r$$$. Find the smallest value of $$$|({a \oplus x}) - ({b \oplus x})|$$$ among all $$$0 \leq x \leq r$$$.

$$$\oplus$$$ is the operation of bitwise XOR, and $$$|y|$$$ is absolute value of $$$y$$$.

Input

The first line contains a single integer $$$t$$$ ($$$1 \le t \le 10^4$$$) — the number of test cases.

Each test case contains integers $$$a$$$, $$$b$$$, $$$r$$$ ($$$0 \le a, b, r \le 10^{18}$$$).

Output

For each test case, output a single number — the smallest possible value.

Example
Input
10
4 6 0
0 3 2
9 6 10
92 256 23
165 839 201
1 14 5
2 7 2
96549 34359 13851
853686404475946 283666553522252166 127929199446003072
735268590557942972 916721749674600979 895150420120690183
Output
2
1
1
164
542
5
3
37102
27934920819538516
104449824168870225
Note

In the first test, when $$$r = 0$$$, then $$$x$$$ is definitely equal to $$$0$$$, so the answer is $$$|{4 \oplus 0} - {6 \oplus 0}| = |4 - 6| = 2$$$.

In the second test:

  • When $$$x = 0$$$, $$$|{0 \oplus 0} - {3 \oplus 0}| = |0 - 3| = 3$$$.
  • When $$$x = 1$$$, $$$|{0 \oplus 1} - {3 \oplus 1}| = |1 - 2| = 1$$$.
  • When $$$x = 2$$$, $$$|{0 \oplus 2} - {3 \oplus 2}| = |2 - 1| = 1$$$.

Therefore, the answer is $$$1$$$.

In the third test, the minimum is achieved when $$$x = 1$$$.