Description:
Define the binary encoding of a finite set of natural numbers $$$T \subseteq \{0,1,2,\ldots\}$$$ as $$$f(T) = \sum\limits_{i \in T} 2^i$$$. For example, $$$f(\{0,2\}) = 2^0 + 2^2 = 5$$$ and $$$f(\{\}) = 0$$$. Notice that $$$f$$$ is a bijection from all such sets to all non-negative integers. As such, $$$f^{-1}$$$ is also defined.
You are given an integer $$$n$$$ along with $$$2^n-1$$$ sets $$$V_1,V_2,\ldots,V_{2^n-1}$$$.
Find all sets $$$S$$$ that satisfy the following constraint:
- $$$S \subseteq \{0,1,\ldots,n-1\}$$$. Note that $$$S$$$ can be empty.
- For all non-empty subsets $$$T \subseteq \{0,1,\ldots,n-1\}$$$, $$$|S \cap T| \in V_{f(T)}$$$.
Due to the large input and output, both input and output will be given in terms of binary encodings of the sets.
Input Format:
The first line of input contains a single integer $$$n$$$ ($$$1 \leq n \leq 20$$$).
The second line of input contains $$$2^n-1$$$ integers $$$v_1,v_2,\ldots,v_{2^n-1}$$$ ($$$0 \leq v_i < 2^{n+1}$$$) — the sets $$$V_i$$$ given in their binary encoding where $$$V_i = f^{-1}(v_i)$$$.
Output Format:
The first line of output should contain an integer $$$k$$$ indicating the number of possible $$$S$$$.
In the following $$$k$$$ lines, you should output $$$f(S)$$$ for all possible $$$S$$$ in increasing order.
Note:
In the first test case, one possible $$$S$$$ is $$$f^{-1}(3) = \{0,1\}$$$. All the non-empty subsets $$$T \subseteq \{0,1,2\}$$$ and the corresponding $$$|S \cap T|$$$, $$$f(T)$$$ and $$$V_f(T)$$$ are as follows:
$$$T$$$$$$|S\cap T|$$$$$$f(T)$$$$$$V_{f(T)}$$$$$$\{0\}$$$$$$1$$$$$$1$$$$$$\{0,1,2,3\}$$$$$$\{1\}$$$$$$1$$$$$$2$$$$$$\{0,1,2,3\}$$$$$$\{2\}$$$$$$0$$$$$$4$$$$$$\{0,1,2,3\}$$$$$$\{0,1\}$$$$$$2$$$$$$3$$$$$$\{0,1,2,3\}$$$$$$\{0,2\}$$$$$$1$$$$$$5$$$$$$\{0,1,2,3\}$$$$$$\{1,2\}$$$$$$1$$$$$$6$$$$$$\{0,1,2,3\}$$$$$$\{0,1,2\}$$$$$$2$$$$$$7$$$$$$\{2,3\}$$$