= Solution
Disconnect the internal \b[Z <alkene> by a non-stabilized <Wittig reaction>]:
$$
\mathrm{MeO_2C(CH_2)_5CH{=}PPh_3}
\quad+\quad
\mathrm{OHCCH(nPr)CH_2CH{=}CH_2}.
$$
The <aldehyde> <carbon> becomes the right-hand internal <alkene> <carbon>; the ylide <carbon> becomes the left-hand one. Both the methyl <ester> and the remote terminal <alkene> survive the olefination.
Make the <aldehyde> from pentanal by <Stork enamine reaction>: form its <pyrrolidine> <enamine>, react at <carbon> with allyl bromide, then hydrolyse to $\mathrm{OHCCH(nPr)CH_2CH{=}CH_2}$. The original alpha <carbon> of pentanal now bears allyl and n-propyl substituents. Control the alkylation to avoid multiple substitutions.
The other branch needs a three-carbon extension of the supplied four-carbon bromo <alcohol>. Protect the OH of $\mathrm{HO(CH_2)_4Br}$ as a <tetrahydropyranyl ether>. Convert the bromide into an <organocuprate> and add it conjugatively to <methyl acrylate>:
$$
\mathrm{THPO(CH_2)_4{-}Cu}
+\mathrm{CH_2{=}CHCO_2Me}
\longrightarrow\mathrm{THPO(CH_2)_6CO_2Me}.
$$
Here the organocopper formula is shorthand for a suitable cuprate preparation, not a claim that an isolated monomer is necessary. The acrylate contributes two methylenes and the <ester> <carbonyl>. Remove the THP group, replace the resulting terminal OH by Br, and react with $\mathrm{PPh_3}$ to form $\mathrm{[Ph_3PCH_2(CH_2)_5CO_2Me]^+Br^-}$. Deprotonate adjacent to <phosphorus> to generate the non-stabilized ylide. This proton is much more acidic than the <ester> alpha protons, but controlled base stoichiometry and low temperature remain important.
Add the allylated <aldehyde> under conditions favouring the usual kinetic \b[<Z-selective Wittig olefination>], and separate any E isomer. The result is
$$
\boxed{(Z)\text{-}\mathrm{MeO_2C(CH_2)_5CH{=}CHCH(nPr)CH_2CH{=}CH_2}.}
$$
The four-carbon starting <alcohol> plus the three-carbon acrylate backbone supplies the seven-carbon acid-derived chain; the <ester> Me gives an eighth <carbon>. Pentanal plus allyl gives the other eight. The total is sixteen <carbons>, including the terminal <alkene>. A two-carbon extension would produce the wrong chain length.
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