PROBLEM STATEMENT

For non-negative integers A and B, let A&B denote the bitwise AND operation.
That is, for each i, the i-th bit of A&B in binary representation is 1 if and only if the i-th 
bits of A and B are 1.



We call a set of non-negative integers S cool if the following conditions are satisfied.


For any two distinct elements A, B in S, A&B > 0.
For any three distinct elements A, B, C in S, A&B&C = 0.



You are given a vector<long long> subset and an int N.
All elements in subset are distinct.
Compute a cool set with N distinct elements such that the cool set contains each element of subset 
and each element is between 1 and 2^60 - 1, inclusive.
Return one such cool set as a vector<long long> with elements in increasing order.
If there are multiple solutions, return the lexicographically smallest one.
If there are no such cool sets, return an empty vector<long long> instead.




DEFINITION
Class:BitwiseAnd
Method:lexSmallest
Parameters:vector<long long>, int
Returns:vector<long long>
Method signature:vector<long long> lexSmallest(vector<long long> subset, int N)


CONSTRAINTS
-N will be between 3 and 50, inclusive.
-subset will contain between 0 and N elements, inclusive.
-Each element of subset will be between 1 and 2^60 - 1, inclusive.
-All the elements in subset will be distinct.
-Elements in subset will be sorted in increasing order.


EXAMPLES

0)
{14, 20}
3

Returns: {14, 18, 20 }

There are several possible cool sets.
For example, the following sets are cool and each of them contains all the elements of subset.

{14, 18, 20}
{14, 20, 26}
{14, 20, 50}
...
  
Among these sets, the first one is the lexicographically smallest one.



1)
{11, 17, 20}
4

Returns: { }

There is no cool set because (11&20) equals 0.

2)
{99, 157}
4

Returns: {99, 157, 262, 296 }



3)
{1152921504606846975}
3

Returns: { }

The element in subset equals to 2^60-1.
Note that each element of your cool set should be less than or equal to 2^60-1. 

4)
{}
5

Returns: {15, 113, 402, 676, 840 }



5)
{1, 3, 5, 7, 9, 11}
6

Returns: { }



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