Archaeologists
Scientists and archaeologists on the planet Olympia have recently uncovered the ruins of an ancient city from a previously unknown civilization. The remnants consist of N towers and M walls, with each wall connecting a specific pair of towers. In a simplified model, towers are represented as points on a plane, and walls are segments connecting these points. Importantly, two walls cannot intersect each other, although they may share a common tower. To explore the city, K archaeologists have been deployed within the city limits. In this model, their positions are also represented as points on a plane. None of the archaeologists are positioned on a tower or wall. Since there is no mobile or radio communication available, archaeologists can only communicate when they meet. Two archaeologists can meet if they can both reach the same point on the plane from their starting positions without crossing any walls or towers. An archaeologist can move from one point to another along any curve.
**Task**
Develop a program that, using the data on the locations of towers, walls, and archaeologists, determines the number of pairs of archaeologists who can meet each other.
**Input**
The first line contains three natural numbers N, M, and K (1 ≤ N ≤ 5∙10^4, 1 ≤ M ≤ 10^5, 1 ≤ K ≤ 5∙10^4) — representing the number of towers, walls, and archaeologists, respectively. The following N lines each contain 2 integers, with absolute values not exceeding 10^6 — these are the coordinates of the towers. No two towers share the same coordinates. The next M lines each contain 2 different integers — the indices of the towers connected by each wall. Towers are numbered from 1 to N. It is guaranteed that two walls do not share any points other than their endpoints. Additionally, no tower lies on any wall, except at the endpoints. The subsequent K lines each contain 2 integers, with absolute values not exceeding 10^6 — these are the coordinates of the archaeologists. No archaeologist is positioned on a wall or tower.
**Output**
The output file, archaeology.sol, should contain a single number — the count of pairs of archaeologists who can meet each other.
**Scoring**
The test set is divided into 6 blocks, each with specific additional conditions:
1. 15 points: 2 ≤ N ≤ 4, 1 ≤ K ≤ 10. 2. 15 points: 3 ≤ N ≤ 1000, 1 ≤ K ≤ 1000, with each tower being the endpoint of no more than two walls. 3. 20 points: All walls are parallel to the coordinate axes, and all coordinates do not exceed 500 in absolute value. 4. 15 points: 2 ≤ N∙K ≤ 1,000,000. 5. 15 points: All walls are parallel to the coordinate axes. 6. 20 points: No additional restrictions.