How well do you know a major root position chord?

This is a tricky question, of course. Everyone knows how to build a major root position chord. (They consist of a major 3rd and minor 3rd).

However, can you play this melody and harmonize it using these chords only with both hands which would be one octave apart?

I can assure you this is easier said than done.

Here is the PDF file for printing out this theme.

And of course, feel free to add long pedal points over these chords on different notes.