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Each visitor to an amusement park needs to buy a ticket. Tickets can be Platinum, Gold, or Economy. Visitors are classified as Old, Middle-aged, or Young. The following facts are known about visitors and ticket sales on a particular day:

- $140$ tickets were sold.
- The number of Middle-aged visitors was twice the number of Old visitors, while the number of Young visitors was twice the number of Middle-aged visitors.
- Young visitors bought $38$ of the $55$ Economy tickets that were sold, and they bought half the total number of Platinum tickets that were sold.
- The number of Gold tickets bought by Old visitors was equal to the number of Economy tickets bought by Old visitors.

Which of the following statements $MUST$ be $FALSE$?

- The numbers of Old and Middle-aged visitors buying Economy tickets were equal
- The numbers of Old and Middle-aged visitors buying Platinum tickets were equal
- The numbers of Middle-aged and Young visitors buying Gold tickets were equal
- The numbers of Gold and Platinum tickets bought by Young visitors were equal