Max and Distances

Max found an array $x$ of $n$ integer points on a line, sorted in non-decreasing order. Max also has two permutations $a$ and $b$ of the numbers from $1$ to $n$, which were given to him as a birthday present.

Max decided to play with them. He built a distance matrix $d$ where $d_{i,j} = |x_{a_i} - x_{b_j}|$. In other words, $d_{i,j}$ is the distance between points $a_i$ and $b_j$.

Before Max could finish playing, Katy cleaned up and $x$, $a$, and $b$ were all lost. Max only has the matrix $d$ left. He wants to restore a non-decreasing array $x$ and two permutations $a$ and $b$ that generate a matrix equal to $d$. However, Max may have miscalculated $d$, in which case $x$, $a$, and $b$ cannot be restored.

Help Max restore them.

The first line contains a natural number $n$ ($1 \le n \le 1000$). The next $n$ lines contain the matrix $d$ ($0 \le d_{i,j} \le 10^9$). All numbers in the matrix are integers.

If there exist $x$, $a$, and $b$ that generate a matrix equal to $d$, print <<YES>> on the first line. On the second line, print the non-decreasing array of integers $x$ ($|x_i| \le 10^9$; $x_i \le x_{i+1}$). On the third line, print the permutation $a$ of the numbers from $1$ to $n$. On the fourth line, print the permutation $b$ of the numbers from $1$ to $n$.

If no such $x$, $a$, and $b$ exist, print <<NO>>.

The input may be large, so it is recommended to use the fast input and output streams of your language.