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

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 $$$v_1,v_2,\ldots,v_{2^n-1}$$$ ($$$0 \leq v_i \lt 2^{n+1}$$$) — 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.

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:

$$$T$$$	$$$\|S\cap T\|$$$	$$$f(T)$$$	$$$V_{f(T)}$$$
$$$\{0\}$$$	$$$1$$$	$$$1$$$	$$$\{0,1,2,3\}$$$
$$$\{1\}$$$	$$$1$$$	$$$2$$$	$$$\{0,1,2,3\}$$$
$$$\{2\}$$$	$$$0$$$	$$$4$$$	$$$\{0,1,2,3\}$$$
$$$\{0,1\}$$$	$$$2$$$	$$$3$$$	$$$\{0,1,2,3\}$$$
$$$\{0,2\}$$$	$$$1$$$	$$$5$$$	$$$\{0,1,2,3\}$$$
$$$\{1,2\}$$$	$$$1$$$	$$$6$$$	$$$\{0,1,2,3\}$$$
$$$\{0,1,2\}$$$	$$$2$$$	$$$7$$$	$$$\{2,3\}$$$

Problemset