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Your passport is only out of your hands as long as they need it to process your new application then they give it back to you until your new passport arrives. Once your new passport arrives in about 10 business days, you give them your old passport with the receipt for your new one and they will punch two holes in it and give it back to you along with your new one. 

If you need to travel within the amount of time required to get your new passport, you can request an emergency passport with proof of travel in the same day and use it until you get your permanent passport.

Visa runs to Singapore are never recommended as mentioned it is the most difficult and expensive place to get a Visa.