Power or XOR?

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 $$$0$$$.