Closing on 1st property with lease expiring in two weeks

[Van Tran]

Hi everyone, we're finally closing on our first property in a couple of days. And the fixed term lease for one unit is expiring in two weeks. There are two individuals living in the unit and one is the seller's son. One of the tenants stated that he was going to leave at the end of the lease, and the other tenant expressed interest in staying. We're not sure if it's the seller's son that's leaving or staying. 

The agents relayed our contact information to the tenant 5 days ago to get in touch with us, but we haven't heard from him yet. We plan to leave a letter of introduction on his door on the day we close with a lease option. We just aren't sure which one yet. The unit is getting rented at below market rate at $500. If we were to rent it to a new tenant, we would try to rent it at $750/month. 

So we're stuck on these options:

1) Would you offer to let him rent the unit for one more month to give him time to find a new place? If so, would you make him fill out a rental application and do a background check for a one month term? This would also require that he put the utilities in his name for one month because they are currently reimbursing the seller for utilities each month.

1a) Would you rent it at the same low rate or raise it $650? I only feel the need to rent it at the same low rate because I feel that he doesn't have much time to find a new place.

2) Would you not offer the one month option and only offer the unit to him for a one year lease at a discounted $600 (because he is an existing tenant). In this scenario we would definitely make him fill out an application and do a background check.

3) Would you treat him as a new tenant and rent it at market rate for a one year term?