Suppose $a$, $b$, and $c$ are real numbers that satisfy the equations:

$$a + b + c = 5$$

and

$$\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{5}.$$ 

Find the greatest possible value of $a^3 + b^3 + c^3$.

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