= Solution
Number the cyclopentenone <carbonyl> <carbon> $C_1$, its <alkene> <carbons> $C_2$ and $C_3$ with OR at $C_3$, and its saturated <carbons> $C_4$ and $C_5$. The <ester> <alkene> has alpha <carbon> $X$, bearing Me and CO2Et, and beta <carbon> $Y$, bearing the protected carbon substituent. \b[<lithium diisopropylamide> generates the ordinary <ketone> <enolate> at the saturated alpha <carbon> $C_5$.] The <acetonide> is retained.
The <carbon> end of this <enolate> attacks <ester> beta <carbon> Y. Its pi system shifts through X onto the <ester> <oxygen>, forming the first new bond $C_5{-}Y$ and an <ester> <enolate> at X. That <enolate> now attacks the original cyclopentenone beta <carbon> $C_3$ intramolecularly, giving the second new bond $X{-}C_3$. The original <enone> <electrons> move onto its <ketone> <oxygen>, and <protonation> of the final <ketone> <enolate> completes the product. The OR <substituent> is retained at the new bridgehead; it need not leave in either <Michael addition>.
The atom map provides a useful structural check. Between bridgeheads $C_3$ and $C_5$, the three paths are $C_3{-}C_2{-}C_1{-}C_5$, $C_3{-}C_4{-}C_5$, and $C_3{-}X{-}Y{-}C_5$. Their numbers of internal atoms are \b[2, 1 and 2], establishing the <norbornane> skeleton. Both original <carbonyls> survive and both initial <alkenes> are consumed. The two successive <conjugate additions> give this atom map without requiring a concerted <cycloaddition>.
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