Mystery number...

During a contest, Benedict, Bob, Brenda, Brian and Billy were asked to each write a number. They decided to write five consecutive positive integers \(p, q, r, s\) and \(t\) respectively such that \(p>q>r>s>t\) according to their order of birth.

Not only was the sum of the five numbers a perfect cube, the sum of Bob, Brenda and Brian's number was a perfect square. Given that all five numbers were greater than 1 but less than 1000, find their sum.

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