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Problem E
Digital Content Protection

Dan is working for a digital content protection company, which is responsible for the content protection of blu-ray discs based on a standard called Anti Content Misuse (ACM).

The ACM standard works as follows. Assume there are $2^ n$ blu-ray drives/players. We represent these $2^ n$ drives as the leaves of a complete binary tree of height $n$, so that each root-to-leaf path consists of $n$ edges. Each node $u$ in this binary tree is assigned an identifier number and contains a random key $k_ u$. The identifier numbers are assigned as follows. The root, $r$, is assigned $1$. In addition, the left and right children of an internal node having number $i$ are assigned numbers $2i$ and $2i+1$, respectively. This scheme assigns a distinct number to each node in the tree. The keys contained in the nodes are unknown to blu-ray users, but they are available to blu-ray drive manufacturers. Each blu-ray player is assigned the identifier number $i$ ($2^ n \le i \le 2^{n+1}-1$) of its corresponding leaf in the tree. A manufacturer of blu-ray drives embeds the keys associated with the nodes in the path from the root to leaf number $i$ in player number $i$.

To encrypt the content of a blu-ray disc, the company in charge creates a random key $k$ called the master key. First, they encrypt $k$ with the key $k_ r$ (recall $r$ is the root node of binary tree) and write it on the disc as a header. Then, they encrypt the content with $k$, and write the encrypted data on the blu-ray disc. A blu-ray drive first decrypts the header using key $k_ r$ embedded in it and recovers the master key $k$ and then, decrypts the content using the key $k$.

Unfortunately, the keys embedded in a set of blu-ray drives, $R$, are exposed by hackers and published on the web. As a result, we cannot encrypt the master key $k$ using any of these exposed keys. For example, since all blu-ray drives contain $k_ r$, the encryption scheme above does not work any more. There is a solution oversaw for this situation in the ACM standard. At the cost of a larger header, the industry can safely encrypt the content of a new blu-ray disc. They carefully choose a subset of unexposed keys $K$ in the binary tree such that all blu-ray drives, except for drives in $R$, have at least one of the keys in $K$. They encrypt the master key $k$ with each key $k’ \in K$ and put the result in the header (i.e., there are $|K|$ ciphertexts in the header). Now, each active blu-ray drive can decrypt at least one of the ciphertexts in the header and can recover the master key $k$. Dan needs your help to determine a subset of keys $K$ with minimum cardinality (which results in the smallest header) given the identifiers of hacked drives.

Input

The input consists of a single test case. A test case consists of two lines. The first line contains two integers $n$ and $|R|$, where $1 \le n \le 62$ and $1 \le |R| \le 1\, 000$. $|R|$ is the cardinality of $R$, the set of exposed drives. The second line contains $|R|$ integers, which are the identifiers of exposed blu-ray drives. You can assume that there is at least one blu-ray drive not hacked.

Output

Display the identifiers of nodes corresponding to the keys in $K$, satisfying the above requirements and having minimum cardinality, in increasing order and separated with single spaces.

Sample Input 1 Sample Output 1
2 1
5
3 4
Sample Input 2 Sample Output 2
3 3
10 11 12
4 7 13
CPU Time limit 1 second
Memory limit 1024 MB
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License Creative Commons License (cc by-sa)

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