Problem G
Find The Root

Full Binary Tree Example

There is a full binary tree $T$ of size $n > 1$. Recall that a full binary tree is a binary tree in which every node has either $0$ or $2$ children.

Let $r$ denote the root of $T$. Note that $T$ qualifies as a full binary tree if and only if it is rooted at $r$. Let $f$ be a function which takes a node $u$ and returns the median of the sizes of the subtrees of $u$ if $T$ were rooted at $u$ instead. Recall that the median of a list of even size is the mean of the 2 middle elements.

For example, in the illustrated example, $r = 1$. $f(1)=3$, since $1$ has two subtrees of sizes $1$ and $5$, the median of which is $3$. $f(5)=1$, since the tree rooted at node $5$ has 3 subtrees, of sizes $1$, $1$, and $4$.

A subset $S = \left\{ u_1, u_2, \dots , u_ m \right\} $ of the nodes of $T$ is chosen with $n / 2 < m \leq n$. You are given the list $[f(u_1), f(u_2), \dots , f(u_ m)]$ and you have to determine whether $r$ is in $S$.

Input

The first line consists of an integer $m$ ($2 \leq m \leq 2 \cdot 10^5$), the size of $S$. The second line consists of $m$ integers, the outputs of $f$ on each node in $S$. It is guaranteed that a valid full binary tree $T$ was used in generating test data for this problem.

Output

Print “yes” if $r$ is in $S$ and “no” if $r$ is not in $S$. If you cannot determine whether $r$ is in $S$ from the given information, print “impossible”.

2
1 2

yes

4
1 2 6 6

no