= Solution
\b[The six carbon environments are assigned as follows], with <chemical shifts> at the centres of the <fluorine>-split multiplets:
$$
\begin{array}{c|r|l}
\text{label}&\delta/\mathrm{ppm}&\text{multiplicity}\\\hline
\mathrm C_e\ (\mathrm{CF_3})&122.11&\text{quartet}\\
\mathrm C_f\ (\mathrm{CF_3CH{=}})&131.335&\text{quartet}\\
\mathrm C_g\ (\mathrm{{=}CHCO_2})&128.97&\text{quartet}\\
\mathrm C_h\ (\mathrm{C{=}O})&163.9&\text{singlet}\\
\mathrm C_j\ (\mathrm{OCH_2})&61.7&\text{singlet}\\
\mathrm C_k\ (\mathrm{CH_3})&14.1&\text{singlet}
\end{array}
$$
The supplied <carbon-13> spectrum is interpreted as <proton-decoupled NMR>, as indicated by its unsplit ethoxy signals. <Proton decoupling> removes C–H splitting but leaves C–F <scalar spin-spin coupling>. Three equivalent spin-$\tfrac12$ <fluorine> nuclei give each resolved quartet its $1:3:3:1$ intensity pattern. The carbonyl's much smaller four-bond C–F interaction is unresolved in the plotted spectrum; “singlet” describes what is observed, rather than proving an exactly zero interaction.
The wide quartet is centred at $(127.48+116.74)/2=122.11$ ppm. The other two centres are $(132.04+130.63)/2=131.335$ and $(129.09+128.85)/2=128.97$ ppm. The signal near $77$ ppm is the <deuterated chloroform> <solvent>, not another carbon of the analyte. The structure labels skip $\mathrm C_i$: the ethoxy methylene is $\mathrm C_j$.
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