Set
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.
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 $V_i = f^{-1}(v_i)$.
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.
Input
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
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.
3
15 15 15 15 15 15 12
5
63 63 63 63 6 63 63 63 63 63 63 5 63 63 63 63 63 63 8 63 63 63 63 2 63 63 63 63 63 63 63
4
3
5
6
7
1
19
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:
| $T$ | $|S\cap T|$ | $f(T)$ | $V_{f(T)}$ |
| $\{0\}$ | $1$ | $1$ | $\{0,1,2,3\}$ |
| $\{1\}$ | $1$ | $2$ | $\{0,1,2,3\}$ |
| $\{2\}$ | $0$ | $4$ | $\{0,1,2,3\}$ |
| $\{0,1\}$ | $2$ | $3$ | $\{0,1,2,3\}$ |
| $\{0,2\}$ | $1$ | $5$ | $\{0,1,2,3\}$ |
| $\{1,2\}$ | $1$ | $6$ | $\{0,1,2,3\}$ |
| $\{0,1,2\}$ | $2$ | $7$ | $\{2,3\}$ |