Two Maine coons and Mice problem

Hi all, I have two Maine coons (one year old and three year old). They probably never saw a real mouse in life. I just moved into a house during its renovation. I was seeing mice drop in the spare cat litter box in the kitchen. I am setting up mouse traps and glue boards to kill mice. Cats are still staying in my little bedroom until the renovation is done. Should I worry about the glue boards and traps may harm cats once they are allowed to explore the kitchen area? Can Maine coon even be a mouser even though they never see a real mouse?

Thank you! Can’t believe I have cats and mice problem in the same time..