Non-Coprime Partition
Description
Find out if it is possible to partition the first $n$ positive integers into two non-empty disjoint sets $S_1$ and $S_2$ such that:
Here $\mathrm{sum}(S)$ denotes the sum of all elements present in set $S$ and $\mathrm{gcd}$ means thegreatest common divisor.
Every integer number from $1$ to $n$ should be present in exactly one of $S_1$ or $S_2$.
The only line of the input contains a single integer $n$ ($1 \le n \le 45\,000$)
If such partition doesn't exist, print "No" (quotes for clarity).
Otherwise, print "Yes" (quotes for clarity), followed by two lines, describing $S_1$ and $S_2$ respectively.
Each set description starts with the set size, followed by the elements of the set in any order. Each set must be non-empty.
If there are multiple possible partitions — print any of them.
Input
The only line of the input contains a single integer $n$ ($1 \le n \le 45\,000$)
Output
If such partition doesn't exist, print "No" (quotes for clarity).
Otherwise, print "Yes" (quotes for clarity), followed by two lines, describing $S_1$ and $S_2$ respectively.
Each set description starts with the set size, followed by the elements of the set in any order. Each set must be non-empty.
If there are multiple possible partitions — print any of them.
Samples
1
No
3
Yes
1 2
2 1 3
Note
In the first example, there is no way to partition a single number into two non-empty sets, hence the answer is "No".
In the second example, the sums of the sets are $2$ and $4$ respectively. The $\mathrm{gcd}(2, 4) = 2 > 1$, hence that is one of the possible answers.