Digital Content Protection

Dan works for a digital content protection company. The company protects the content of blu-ray discs under a standard called Anti Content Misuse (ACM).

The ACM standard works as follows. Suppose there are $2^n$ blu-ray players. Each player is one leaf of a complete binary tree of height $n$, so every root to leaf path uses exactly $n$ edges. Every node $u$ of the tree carries an identifier number and holds a random key $k_u$. Identifiers are assigned this way. The root $r$ gets $1$, and an internal node numbered $i$ has left child $2i$ and right child $2i+1$. This gives every node a distinct number. Blu-ray users never see the keys inside the nodes, but the manufacturers know them. Each player is identified by the number $i$ ($2^n \le i \le 2^{n+1}-1$) of its leaf, and the manufacturer embeds in player $i$ every key on the path from the root to leaf $i$.

To encrypt a disc, the company generates a random key $k$ called the master key. It encrypts $k$ with the root key $k_r$ and writes the result on the disc as a header, then encrypts the content with $k$ and writes the encrypted data on the disc. A player decrypts the header with its own copy of $k_r$, recovers the master key $k$, and decrypts the content with $k$.

Hackers extracted the keys embedded in a set $R$ of players and published them on the web, so none of those keys can protect the master key any more. Every player holds $k_r$, so the scheme above is broken. The ACM standard covers this case. The header grows, but the content of a new disc stays safe. The company picks a set $K$ of keys that were not exposed, such that every player outside $R$ holds at least one key of $K$. It encrypts the master key $k$ separately with each key $k' \in K$ and puts the $|K|$ ciphertexts in the header. Every player that was not hacked then decrypts at least one ciphertext of the header and recovers $k$. The smaller $|K|$ is, the shorter the header. Given the identifiers of the hacked players, help Dan find a smallest $K$.

The input is a single test case on two lines. The first line contains two integers $n$ and $|R|$ ($1 \le n \le 62$, $1 \le |R| \le 1000$), where $|R|$ is the size of the set $R$ of exposed players. The second line contains the $|R|$ identifiers of the exposed players. At least one player was not hacked.

For a set $K$ that meets the requirements above and has minimum size, print the identifiers of the nodes holding the keys of $K$ in increasing order, separated by single spaces. Exactly one set of minimum size meets the requirements.