A large vessel containing $100$ kg of a $60\%$ salt solution is filled with $10\%$ salt solution at a rate of $10$ kg$/$min$.$ The solution is removed for the vessel at a rate of $20$ kg$/$min$.$ Assuming complete mixing, find the kilogram of salt in the tank after $5$ minutes.

Amount of salt in the vessel after $5$ minutes $=\_\_\_\_\_\_\_\_\_\_\_\_\_\_$kg