Gerund

Hi,
Are these sentences correct?
1- The best thing to do is to write a note, and asking him to authorize me.
2- The best thing to do is to write a note, and to ask him to authorize me.
But I think that no. 2 is a parallel structure and “to ask” is correct. But I am not sure about no. 1. If it is correct why gerund (“asking”) is used after “and”. What is this structure?

Regards