At a birthday party, there were $\frac{1}{4}$ as many red balloons as green balloons and $\frac{3}{5}$ as many red balloons as purple balloons. Each red balloon was $4$ grams, each green balloon was $5$ grams and each purple balloon was $7$ grams. The total mass of the balloons was $642$ grams. How many purple balloons were there?~\uline{~~~~~~~~~~}~ 
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