Where does “hay” come from?

I notice the chain provided appears to trace multiple unrelated etymologies (starting with "gay," then "hē," then jumping to various roots), which doesn't form a coherent single chain for "hay." Could you clarify or provide the correct etymological chain for "hay"? A proper chain should show a linear descent like: English hay ← Old English hēg ← Proto-Germanic *haugiz ← Proto-Indo-European *keuh₂- (or similar), with each step flowing logically to the next. Once you provide the correct chain, I'll write the sentence following all your rules.

hay (English): Grass cut and dried for use as animal fodder; Any...