5408：CRB and Farm

CRB has a farm. His farm has a shape of convex polygon. There are $K$ barns in the farm. All of them are strictly inside the polygon. CRB wants to build a fence enclosing all the barns. He prepared $2K$ posts and steel strips of exactly the same length with the perimeter of his farm. A post can only be placed on the vertex of the polygon(farm), and steel strips must enclose all posts. Now, CRB wonders whether he can build a fence with currently available resources or not. Can you help him?

There are multiple test cases. The first line of input contains an integer $T$, indicating the number of test cases. For each test case:
The first line contains an integer $N$ – the number of vertices of the convex polygon(the shape of CRB‘s farm).
Then $N$ lines follow, $i$-th line containing two integers $x$ and $y$, the coordinates of $i$-th vertex. The vertices are given in counter-clockwise order.
The first line contains an integer $K$ – the number of barns.
Each of the next $K$ lines contains two integers $x$ and $y$, the coordinates of the barn.
1 ≤ $T$ ≤ 9
3 ≤ $N$, $K$ ≤ 2 * $10^{5}$
$-10^{9}$ ≤ $x$, $y$ ≤ $10^{9}$
All points(including farm vertices and barns) are pairwise different.
It is guaranteed that the barns never all lie on a line.

For each test case, output “Yes” if it is possible to build a fence, otherwise output “No”.
If possible, output a solution in the following format:
On the first line, output an integer $M$ - the number of posts used to build a fence.
On the next line, you should output the indices of the vertices(1-based) where you place posts in increasing order.
Your solution will be accepted if $M$ ≤ $2K$ and the length of the fence is not longer than the perimeter of the farm. Also, the barns must be strictly inside the fence.