If you're able to acquire an Tokkul-Zo instead of an archer's rings: it's part of OSRS gold the older kiln quest and has +4 in all combat stats , excluding prayer and strength, which makes it more powerful than your archer's jewelry. A ring amulet is probably a better thing to purchase rather than a fury. It offers a greater ranged attack bonus although it defends you slightly less well and has the prayer reward removed.
There's no need for a robin cap or hood, but you will need an archer helm. For a quick stat check The archer helm gives superior defense, however the robin hat offers +2 more defensive range. It's up to you, the differences are small. Robins work well as a cosmetic item and can be customized to earn loyalty points at Xuan in burthorpe and Varrock, but.
If you don't already have them, you need to purchase some barrow's gloves. They're extremely strong for the glove slot and extremely helpful because of it. It is also recommended to buy some ranging boots The best option is snakeskin in order to avoid ranger boots, since they (12M) aren't within your budget.
The shield slot will likely involve an illuminated god book. Zamorak's book is priced at 22 million it's a better alternative is armadyl's book of law with a price of 1.7M and it has 2 less ranged offence, however, it does have defence capabilities, and when you've illuminated it, you'll earn a big prayer bonus.
The cape you wear should be one of your own unless doing something else that requires it to work, such as shooting ogres within a cage. In this case, you must wear your finest ardougne-style cloak or something else with offensive/prayer bonuses. Red dragonhide, obviously and unless you're looking to cheap RuneScape gold gain rid of. I hope this can help!
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