An amulet of ranging is probably a better item to purchase than a fury: it has a higher ranged attack bonus , but it protects you only a bit less well, and has none of the prayers bonus.
You won't need a Robin Hood hat or an archer helm for a quick stat check, the archer helm provides higher defence but the robin's hat gives you +2 more ranged offense. It's really your call but the differences are not that big. Robins are great as a decorative item. It is recolored to earn loyalty points at Xuan in burthorpe and varrock, though.
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