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Two pipes P and Q can fill a tank in $6$ hours and $9$ hours respectively, while a third pipe R can empty the tank in $12$ hours. Initially, P and R are open for $4$ hours, Then P is closed and Q is opened. After $6$ more hours R is closed. The total time taken to fill the tank (in hours) is ____

- $13.50$
- $14.50$
- $15.50$
- $16.50$

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