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Progression of chords sounded relatively familiar.
Two songs came to mind which resembled.
Samples were taken, of the 1993 Sting song "Shape of My Heart" as well as the 1973 Aerosmith song "Dream On" (a sample more heavily relied on by Eminem in the song "Sing for the Moment"). These two samples together, however, fit perfectly; particularly in tone, when properly adjusted for tempo.
They've both been slowed down by a few tones so that they match the slow pace of the Del Rey lyrics. That tempo is about 69bpm. You can test BPM's here on the Space Pirate Online BPM Calculator. Here's the song in its current form on Soundcloud: