A Simple Synthesis of the Lignan Carpanone

The lignan carpanone 1 would present a formidable synthetic challenge if it were to be constructed from simple starting materials in a linear, stepwise manner. It can, however, be assembled rapidly by the following sequence of four simple reactions (none of the intermediates need be rigorously purified) (i) alkylation of 3,4-methylenedioxyphenol with allyl 

Elucidate the reaction sequence and give a mechanistic explanation for the conversion of C into carpanone 1. 

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