A factory has a $20\%$ sugar solution and a $50\%$ sugar solution. The manager needs to produce $100$ kilograms of a $30\%$ sugar solution. How many kilograms of each solution should be mixed to achieve this concentration? Additionally, after producing the $30\%$ sugar solution, the manager needs to dilute it to a $25\%$ sugar solution by adding water. How many kilograms of water should be added? Let's think step by step and output the final answer within \boxed{}.
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