In a forest, there are five trees labeled $A$, $B$, $C$, $D$, and $E$ that are aligned in that order on a straight line. 

- At the midpoint of segment $AB$, there is a daisy.
- At the midpoint of segment $BC$, there is a rose bush.
- At the midpoint of segment $CD$, there is a jasmine.
- At the midpoint of segment $DE$, there is a carnation.

The distance between tree $A$ and tree $E$ is $28$ meters. Additionally, the distance between the daisy and the carnation is $20$ meters. Calculate the distance between the rose bush and the jasmine.

Let's think step by step and output the final answer within \boxed{}.
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