$\textbf{Medalists}$

Five runners, $A$, $B$, $C$, $D$, and $E$, enter the final. The fastest three win a gold, silver, and bronze medal, respectively. The other two get nothing. Who are the three medalists if all of the following statements are false?

• $A$ does not win the gold and $B$ does not get the silver.
• $B$ does not get the bronze and $D$ does not win silver.
• $C$ wins a medal, but $D$ does not.
• $A$ wins a medal, but $C$ does not.
• Both $D$ and $E$ win a medal.