Candles and Shadows

A red convex polygon with $N$ vertices $(A_1, B_1), (A_2, B_2), \cdots, (A_N, B_N)$ and a blue convex polygon with $M$ vertices $(C_1, D_1), (C_2, D_2), \cdots, (C_M, D_M)$ are given. As the figure shows, the blue polygon lies completely inside the red polygon.

Figure 1. A red polygon containing a blue polygon

Let $S$ be the region inside the red polygon that is not inside or on the boundary of the blue polygon. As the figure shows, placing a candle at a point of $S$ lights the yellow region, and the gray region casts a shadow.

Figure 2. The lit region and the shadow region

Given the polygons and the position of the candle, write a program that computes the area of the shadow region.

The first line contains three positive integers $N$, $M$, and $Q$, separated by spaces. They are the number of vertices of the red polygon, the number of vertices of the blue polygon, and the number of queries.

The next $N$ lines give the vertices of the red polygon in counterclockwise order. Line $(i+1)$ contains two integers $A_i$ and $B_i$ separated by a space ($1 \le i \le N$).

Starting from line $(N+2)$, the next $M$ lines give the vertices of the blue polygon in counterclockwise order. Line $(N+i+1)$ contains two integers $C_i$ and $D_i$ separated by a space ($1 \le i \le M$).

Starting from line $(N+M+2)$, the next $Q$ lines describe the queries. Line $(N+M+i+1)$ contains two integers $X_i$ and $Y_i$, meaning that in the $i$-th query the candle is placed at $(X_i, Y_i)$. All $Q$ queries are independent of each other.

Print the answer to each query on its own line, in order.

If the candle is outside the red polygon or on its boundary, print OUT.

If the candle is inside the blue polygon or on its boundary, print IN.

Otherwise, print the area of the shadow region. An absolute or relative error up to $10^{-6}$ is allowed.

All input data satisfy the following conditions.

• $3 \le N \le 10^5$
• $3 \le M \le 10^5$
• $1 \le Q \le 10^5$
• $-10^6 \le A_i \le 10^6$ ($1 \le i \le N$)
• $-10^6 \le B_i \le 10^6$ ($1 \le i \le N$)
• $-10^6 \le C_i \le 10^6$ ($1 \le i \le M$)
• $-10^6 \le D_i \le 10^6$ ($1 \le i \le M$)
• $-10^6 \le X_i \le 10^6$ ($1 \le i \le Q$)
• $-10^6 \le Y_i \le 10^6$ ($1 \le i \le Q$)