The question given above appeared in the 2016 Indian National Chemistry Olympiad (INChO). What I think is that the epoxide ring is being opened by a base-catalysed mechanism ($\mathrm{S_N2}$). I proposed the following few steps for this:
The diisopropylamide ion, which is a very strong base, would not be a good leaving group. To obtain the desired products, how could the mechanism then proceed? And if not, then what would be the real mechanism? I did not find any reliable information on the Internet regarding ring opening using LDA.