$(30$ points total$)$ You are traveling with Cersei Lannister towards Kings Landing and she wishes to test your engineering knowledge. She reminds you that "When you play the Game of Thrones, you win or you die...there is no middle ground." Cersei tells you that a total of $15$ gallons of purified water will be needed to complete your current journey. You find a stream containing $33wt\%$ of poisonous bugs that ends up being your only source of drinking water. Cersei's illegitimate brother Tyrion Lannister has provided you with $18\mathrm{.}5kg$ of a special serum which is capable of removing $99\mathrm{.}999\%$ of the mass of bugs making it safe for drinking. Fearing you need to beat Cersei at her game to stay alive, calculate the volume $($in gallons$)$ of bug$-$filled water necessary for you to achieve the required amount of purified water AND the weight fraction of bugs that ends up in the potion. Note that the serum will NOT mix with the water so the bug$-$filled water it in your process flow diagram. You may also assume that the density of