You are an expert competitive programmer. You will be given a problem statement, test case constraints and example test inputs and outputs. Please reason step by step about the solution (that must respect memory and time limits), then provide a complete implementation in c++17.

Your solution must read input from standard input (cin), write output to standard output (cout).
Do not include any debug prints or additional output.

Put your final solution within a single code block:
```cpp
<your code here>
```

Execution time limit: 2.0 seconds
Memory limit: 512.0 MB

# Problem
The symbol $$$\wedge$$$ is quite ambiguous, especially when used without context. Sometimes it is used to denote a power ($$$a\wedge b = a^b$$$) and sometimes it is used to denote the XOR operation ($$$a\wedge b=a\oplus b$$$).

You have an ambiguous expression $$$E=A_1\wedge A_2\wedge A_3\wedge\ldots\wedge A_n$$$. You can replace each $$$\wedge$$$ symbol with either a $$$\texttt{Power}$$$ operation or a $$$\texttt{XOR}$$$ operation to get an unambiguous expression $$$E'$$$.

The value of this expression $$$E'$$$ is determined according to the following rules:

- All $$$\texttt{Power}$$$ operations are performed before any $$$\texttt{XOR}$$$ operation. In other words, the $$$\texttt{Power}$$$ operation takes precedence over $$$\texttt{XOR}$$$ operation. For example, $$$4\;\texttt{XOR}\;6\;\texttt{Power}\;2=4\oplus (6^2)=4\oplus 36=32$$$.
- Consecutive powers are calculated from left to right. For example, $$$2\;\texttt{Power}\;3 \;\texttt{Power}\;4 = (2^3)^4 = 8^4 = 4096$$$.

You are given an array $$$B$$$ of length $$$n$$$ and an integer $$$k$$$. The array $$$A$$$ is given by $$$A_i=2^{B_i}$$$ and the expression $$$E$$$ is given by $$$E=A_1\wedge A_2\wedge A_3\wedge\ldots\wedge A_n$$$. You need to find the XOR of the values of all possible unambiguous expressions $$$E'$$$ which can be obtained from $$$E$$$ and has at least $$$k$$$ $$$\wedge$$$ symbols used as $$$\texttt{XOR}$$$ operation. Since the answer can be very large, you need to find it modulo $$$2^{2^{20}}$$$. Since this number can also be very large, you need to print its binary representation without leading zeroes. If the answer is equal to $$$0$$$, print $$$0$$$.

## Input Format
The first line of input contains two integers $$$n$$$ and $$$k$$$ $$$(1\leq n\leq 2^{20}, 0\leq k < n)$$$.

The second line of input contains $$$n$$$ integers $$$B_1,B_2,\ldots,B_n$$$ $$$(1\leq B_i < 2^{20})$$$.

## Output Format
Print a single line containing a binary string without leading zeroes denoting the answer to the problem. If the answer is equal to $$$0$$$, print $$$0$$$.

## Examples
```input
3 2
3 1 2
```
```output
1110
```
-----
```input
3 1
3 1 2
```
```output
1010010
```
-----
```input
3 0
3 1 2
```
```output
1000000000000000001010010
```
-----
```input
2 1
1 1
```
```output
0
```

## Note
For each of the testcases $$$1$$$ to $$$3$$$, $$$A = \{2^3,2^1,2^2\} = \{8,2,4\}$$$ and $$$E=8\wedge 2\wedge 4$$$.

For the first testcase, there is only one possible valid unambiguous expression $$$E' = 8\oplus 2\oplus 4 = 14 = (1110)_2$$$.

For the second testcase, there are three possible valid unambiguous expressions $$$E'$$$:

- $$$8\oplus 2\oplus 4 = 14$$$
- $$$8^2\oplus 4 = 64\oplus 4= 68$$$
- $$$8\oplus 2^4 = 8\oplus 16= 24$$$

For the third testcase, there are four possible valid unambiguous expressions $$$E'$$$:

- $$$8\oplus 2\oplus 4 = 14$$$
- $$$8^2\oplus 4 = 64\oplus 4= 68$$$
- $$$8\oplus 2^4 = 8\oplus 16= 24$$$
- $$$(8^2)^4 = 64^4 = 2^{24} = 16777216$$$

For the fourth testcase, $$$A=\{2,2\}$$$ and $$$E=2\wedge 2$$$. The only possible valid unambiguous expression $$$E' = 2\oplus 2 = 0 = (0)_2$$$.

Now solve the problem and return the code.