= Solution
Open the cyclic <ester> at its acyl–<oxygen> bond. The resulting delta-hydroxy acid is
$$
\mathrm{HO_2C-CH(nPr)-CH_2-CH(iPr)-CH(OH)Ph}.
$$
Oxidation-state reversal at the benzylic <alcohol> gives $\mathrm{HO_2C-CH(nPr)-CH_2-CH(iPr)-C(=O)Ph}$. This identifies the key \b[Michael disconnection between the central $\mathrm{CH_2}$ and $\mathrm{CH(iPr)}$ groups]:
$$
\mathrm{PhCOCH_2iPr}+\mathrm{CH_2=C(nPr)CO_2Et}
\longrightarrow\mathrm{PhCOCH(iPr)CH_2CH(nPr)CO_2Et}.
$$
The <nucleophile> is the <enolate> or <enamine> of the phenyl <ketone>. The <electrophile>'s propyl substituent must be alpha to its <ester> <carbonyl>; using a beta-propyl acrylate would move the substituent to the wrong ring position.
Disconnect $\mathrm{PhCOCH_2iPr}$ further to the indicated \b[<acetophenone>]. Crossed aldol addition to acetone followed by dehydration gives $\mathrm{PhCOCH=CMe_2}$; selective conjugate reduction or controlled alkene <hydrogenation> then gives $\mathrm{PhCOCH_2CHMe_2}$. This avoids requiring an efficient SN2 alkylation with a secondary isopropyl halide. The alpha-propyl acrylate can be obtained by alpha-hydroxymethylation/dehydration of <ethyl pentanoate>, $\mathrm{EtO_2CCH_2nPr}$, using formaldehyde.
In the forward direction, perform the Michael coupling, reduce the phenyl <ketone> selectively with <sodium borohydride> while retaining the ordinary <ester>, hydrolyse the <ester>, and cyclize the delta-hydroxy acid by controlled acid-catalysed <lactonization> or an activated-acid coupling. The six-membered ring then has the order \b[<carbonyl>–$\mathrm{CH(nPr)}$–$\mathrm{CH_2}$–$\mathrm{CH(iPr)}$–$\mathrm{CH(Ph)}$–O]. This atom order checks all three substituent positions. No relative configuration is specified, so the scheme does not imply a single <diastereomer> from the <Michael addition> or reduction.
\Image[/past-exam-of-the-chemistry-course-of-the-year-3-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-1-q7-lactone-route.png]
{title=Original lactone-opening and Michael disconnections, with propyl and isopropyl positions tracked separately}
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