Define a function $f(x)$ as the number of distinct elements in sequence $x$. For example:

• $f([1, 2, 3, 3]) = 3$, because the sequence contains three distinct elements: $1, 2, 3$.
• $f([2, 2, 2, 2]) = 1$, because all elements are the same, so there is only one distinct element.

Furthermore, we define a function $g(y)$ as the sum of $f(x)$ for all non-empty subsequences $x$ of sequence $y$, that is:

$$g(y) = \sum_{x \subseteq y, x \neq \emptyset} f(x)$$

In other words, $g(y)$ calculates the total number of distinct elements across all non-empty subsequences of $y$.

A subsequence of a sequence is obtained by deleting zero or more elements from the original sequence while maintaining the relative order of the remaining elements.

Given a positive integer $x$, your task is to construct a sequence $a$ such that $g(a) = x$, or report that no such sequence exists.

Input

The input contains a single positive integer $x$ ($1 \le x \le 10^{18}$).

Output

If no such sequence exists, output a single line containing No.

Otherwise, output Yes on the first line, an integer $n$ representing the length of the sequence on the second line, and $n$ positive integers $a_1, a_2, \dots, a_n$ representing the sequence on the third line.

The constraints are $1 \le n \le 10^5$ and $1 \le a_i \le n$. It can be proven that if a solution exists, there must exist a solution satisfying these constraints.

Examples

Input 1

1

Output 1

Yes
1
1

Input 2

4

Output 2

Yes
2
2 1

Input 3

10

Output 3

Yes
3
3 1 3