Crash and Go(relians)

The Gorelians are a warlike race who travel the universe, conquering new worlds for entertainment. Usually, their space battles are one-sided, but sometimes, even the Gorelians experience defeat. In one such battle, their spaceship was so badly damaged that they had to evacuate to the planet below. Due to the chaos (and the inaccuracy of escape pods), the Gorelians ended up scattered across a large area of the planet (but still small enough that its surface can be modeled as a plane, not a sphere). Your task is to track their efforts to regroup. Fortunately, each escape pod is equipped with a locator, informing its Gorelian of their current coordinates on the planet, and a radio for communicating with other Gorelians. Unfortunately, the radio's range is limited by the available power supply.

When a Gorelian lands on an alien planet, the first thing they do is check their radio to see if they can contact any other Gorelians. If so, they arrange a rendezvous point and gather at that spot. Once together, they pool the power supplies of their radios, increasing their range. Then they repeat the process—checking whom they can reach, arranging to meet, pooling their radios—until they can no longer contact any other Gorelians.

Gorelian technology allows two-way communication if at least one has a radio with sufficient range to cover the distance between them. For example, if Alice has a radio with range $40$ km and Bob has one with $30$ km, but they are $45$ km apart (Figure $1$), they cannot communicate. However, if the distance between them is $35$ km (Figure $2$), Alice's radio can reach Bob, and they can communicate.

If a Gorelian successfully connects with others, they meet at a location which is the average of their positions. In the case of Alice and Bob, this is the midpoint between $A$ and $B$ (Figure $3$). Gorelians switch off their radios while traveling; they will not attempt to communicate with others until everyone is assembled at the meeting spot. When Gorelians gather, they combine their radios, creating a new one with greater range. In particular, the area covered by the new radio equals the sum of the areas covered by the old radios. For example, if Alice's range is $40$ km, her radio covers an area of $1600\pi$ km^2. Bob's radio covers $900\pi$ km^2. By merging them, they cover $2500\pi$ km^2, i.e., they have a range of $50$ km. They then attempt to reach other Gorelians again.

This process continues until no further contacts can be made. As an example, suppose the following Gorelians have landed, each with a radio radius of $30$ km: Alice $(100,100)$, Bob $(130,80)$, Cathy $(80,60)$, and Dave $(120,150)$. None of them can communicate with another (Figure $5$). Then Eddie lands at $(90,80)$ (Figure $6$). Eddie can contact Alice and Cathy, so they arrange to meet at $(90,80)$, the average of their locations. Combining their radios gives them a range of \sqrt{2700} \approx 51.96 km (Figure $7$).

Now, with their new range, they check again and find they can reach Bob. They meet Bob at $(110,80)$ and combine radios, obtaining a radius of $60$ (Figure $8$). Unfortunately, this is still not enough to reach Dave, so Dave remains isolated.