Challenge: A geometric sequence has a difference of $24$ between its first two terms, and a difference of $648$ between its fourth and fifth terms. Find the starting value of the sequence and the ratio between terms.

1. Find $a$ and $r$, assuming that the initial value $a$ and the common ratio $r$ are both positive.

   Hint (a):

   $ar − a = 24$
   $ar^{4} − ar^{3} = 648$

2. Extra Challenge: Assuming $a$ and $r$ could be positive or negative, there are FOUR solutions. Can you find them all?