410467: GYM104023 L Novice Magician

HoshiYo the Fox is a magician who is new to the magic school. As a novice, he is not proficient in many kinds of magic, especially those about numbers.

One day, HoshiYo learns a kind of magic that can change the integers in an array of length $$$2^n$$$. However, due to a lack of proficiency, he can only change half of the integers in the array each time he uses magic. Formally, let's denote the array as $$$a_0, a_1, \dots, a_{2^n-1}$$$, the following action can be performed by using the magic once:

• Choose $$$2^{n-1}$$$ different integers in the array and an integer $$$x$$$ (possibly negative), add $$$x$$$ to one of them, add $$$x+2$$$ to another one of them, add $$$x+4$$$ to another one of them, ..., and add $$$x+2^n-2$$$ to the last remaining one. In other words, choose $$$2^{n-1}$$$ different indices ranging from $$$0$$$ to $$$2^n-1$$$ denoted by $$$p_0,p_1,\dots,p_{2^{n-1}-1}$$$, and add $$$x+2i$$$ to $$$a_{p_i}$$$ for each integer $$$i$$$ within $$$0 \le i < 2^{n-1}$$$.

Initially, each integer in the array is $$$0$$$. HoshiYo has an ideal array $$$b_0, b_1, \dots, b_{2^n-1}$$$ in his mind. He wonders if he can get this ideal array by using magic at most $$$2^n$$$ times, i.e., to make $$$a_0=b_0,a_1=b_1,\dots,a_{2^n-1}=b_{2^n-1}$$$.

The first line contains an integer $$$n$$$ ($$$1 \le n \le 11$$$), indicating that the length of the array is $$$2^n$$$.

The second line contains $$$2^n$$$ integers $$$b_0, b_1, \dots, b_{2^n-1}$$$ ($$$0 \le b_i \le 10^5$$$), indicating the ideal array.

If the ideal array cannot be obtained by using magic at most $$$2^n$$$ times, output NO in a single line.

Otherwise, output YES in the first line and an integer $$$k$$$ ($$$0 \le k \le 2^n$$$) in the second line indicating the number of times to use magic. Then output $$$k$$$ lines, each of which contains $$$2^{n-1}+1$$$ integers $$$x, p_0, p_1, \dots, p_{2^{n-1}-1}$$$ separated by spaces. You need to ensure that $$$p_0, p_1, \dots, p_{2^{n-1}-1}$$$ are distinct integers between $$$0$$$ and $$$2^n-1$$$, and each of $$$a_0, a_1, \dots, a_{2^n-1}$$$ is always in the range of $$$[-10^{18},10^{18}]$$$.

If there are multiple solutions, output any.

2
1 14 5 14

YES
4
0 1 0
2 1 2
12 1 3
-1 0 2

2
11 45 1 4

NO

For the first sample, the result after each use of magic is as follows:

1. $$$\{0+2,0+0,0,0\}=\{2,0,0,0\}$$$
2. $$$\{2,0+2,0+4,0\}=\{2,2,4,0\}$$$
3. $$$\{2,2+12,4,0+14\}=\{2,14,4,14\}$$$
4. $$$\{2-1,14,4+1,14\}=\{1,14,5,14\}$$$