1st time getting rid of a Tenant

[Abel Rodrigues]

For reference property is in Waterbury, CT

I have a Really bad inherited tenant that is on a m-m lease and I FINALLY ask them to leave but have not done an official eviction yet. I want to see how this goes as far as not renewing their lease. i want to serve them with a "Notice to Vacate" we do not have anything in the lease stating minimum termination period and in CT it looks like there is no minimum termination period. though the min in 0 day in CT I'm going to give them until the end on September (55 days) to get out. I just want to ask people experience with a notice to vacate, do they get out on their own or do they milk it until they get evicted so they can stay longer? is 55 days too long?

Also the tenant wanted me to give him a "notice to quit", why? i feel like they are trying to play a legal game.