Where does “metabolism” come from?

I need to work through this etymology chain carefully, but I notice the chain provided appears incomplete or problematic—it shows Ancient Greek βάλλω twice and doesn't include the "meta-" prefix component that would be essential to "metabolism." However, following your instruction to "state only what the chain supports" and not invent intermediate steps: I cannot write an accurate sentence because the chain as given doesn't contain the necessary etymological path for "metabolism." The word requires both "meta-" (from Ancient Greek μετά) and a root related to "throwing" or "change," but the provided chain lacks the complete derivation. Could you verify the etymology chain? A proper metabolism etymology should trace through: - English metabolism - from Ancient Greek metabolē (change) - from metabollein (to change) - from meta- + ballein (to throw) - from Ancient Greek βάλλω and related elements

metabolism (English): The chemical processes that occur within a living...