First, if you have an attorney representing you in the bankruptcy do not do anything but give him the package and go from there. If you are not represented then you have to decide if you want to continue with the bankruptcy, and the reasons behind the bankruptcy. If you are 3 months behind on your mortgage, you are not going to be able to keep your home in a Chapter 7. In order to reaffirm the home you must be current on the payments, otherwise you surrender the house, or they proceed with mortgage foreclosure.
If your primary goal is to keep the home and you have no other issues with regard to debt you may want to consider dimissing the Chapter 7 and accepting the new loan modification package. They will not foreclose if you accept their package, but make sure of that and get it in writing.
I am curious how you would receive a Motion to Lift Stay the same day you filed bankruptcy, they wouldn't have had time to issue that Motion I wouldn't think. However, this is a standard motion because a Chapter 7 only delays the inevitable, it does not stop a foreclosure, it merely puts a stay in place. The mortgage company always motions for a lift and its almost always granted since the debtor is behind on payments.
If you accept the loan modification plan you will have to make sure that they will accept a reaffirmation if you continue with the bankruptcy and you will have to amend your petition to show the reaffirmation, and new loan payments. This way you may be able to proceed with the bankruptcy and keep your home out of it. If you do not have an attorney helping you it may be very wise to seek a consultation on this matter with one in your local area.