Problem $3.(30$ points$)$ "Per aspera ad astra," through adversity to the stars, is the Stevens motto. Our classmates $-$ the self$-$proclaimed math genius Lem E$.$ Hackett and his buddy Ben Bitdiddle got a head start planning their senior design project way back! They have been busily scanning the heavens for alien life every night since kindergarten and, despite any light pollution $($or probably because of it$),$ have already made the following discoveries that have not yet been announced publicly:

Five$-$headed aliens from the Sombrero galaxy arrive on Earth every $3$ years; their most recent arrival was in $2021.$

Three$-$legged aliens from the Triangulum galaxy arrive on Earth every $7$ years; their most recent arrival was in $2019.$

Seven$-$eyed aliens from the Eyes of Sauron galaxy arrive on Earth every $13$ years; their latest arrival was in $2020.$

Lem and Ben plan to record the next trifecta $-$ the arrival of all three alien forms in the same year $-$ and to announce their breakthrough senior design project results at their graduation ceremony.

When will Lem and Ben graduate? Formulate your problem precisely, and show and explain all steps in your solution, using only techniques presented in class.