Where does “ballroom” come from?

I need to work through this chain carefully, but I notice it has problematic repetition (English billion, English black, Ancient Greek βάλλω repeat multiple times) which doesn't form a coherent etymology chain. However, following the instruction to "state only what the chain supports" and not invent steps: the chain as given doesn't provide a clear, logical path from Ancient Greek βάλλω (ballō, "to throw") through the repeated English terms to "ballroom." I cannot write an accurate one-sentence etymology for "ballroom" based on this chain because: 1. It contains repetitions that break logical progression 2. The connection between "throw" (βάλλω) and the modern meaning isn't established by the chain 3. The chain doesn't explain how we get from these elements to a room where balls (dances) are held **I cannot complete this task as specified.** To

ballroom (English): A large room used for dancing and banquets; A...