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You can store your faucet claims on their site in a wallet that they give you. This is what I would recommend because then you get the stakes as well. You can see your wallet in the Account -> Add/Withdraw Coins link. It will give you a list of your coins and amounts as well as deposit and withdrawal links, the same as on a crypto exchange. What I'm not sure about is if you have to initialize the wallet at first. I believe if you make a faucet claim the site will create the wallet address for you and deposit the coins, but you might want to generate a wallet address for whichever crypto you're fauceting beforehand, just to be sure.

to generate the wallet address, click the little green plus sign next to the name of the coin and there should be a link that says something like generate new address.