Where does “firehouse” come from?

I need to work through this chain carefully, but I notice there's a significant problem: the etymological chain provided doesn't actually lead to "firehouse." The chain traces back through "house" and various Proto-Indo-European roots, but "firehouse" is a compound word (fire + house), and the chain only accounts for the "house" portion. Additionally, the chain appears corrupted or incomplete—it shows "Proto-Indo-European éǵh₂" twice and ends with disconnected elements that don't form a coherent etymological path. Given the rules require me to "state only what the chain supports" and "do not invent," I cannot write an accurate sentence based on this chain as presented. Could you provide a corrected etymological chain for "firehouse"? I would need either: 1. A proper chain showing how "firehouse" as a complete compound developed, or 2. Clar

firehouse (English): A house containing a fire to heat it; a...