Answer ALL parts

Molecule $x,$ molecule $Y$ and molecule $Z$ have the related structures shown below:

Spectra for each of the molecules have been recorded and are presented in the accompanying analytical data booklet as three datasets: dataset $1,$ dataset $2$ and dataset $3.$ Gas phase IR spectra are shown on the left, and electron impact ionization mass spectra on the right.

Molecule $x$ has the composition $65\mathrm{.}1\%$ carbon, $28\mathrm{.}9\%$ oxygen and $6\mathrm{.}0\%$ hydrogen.

$($a$)$ Calculate the atom ratio carbon:oxygen:hydrogen in molecule $x,$ and suggest a complete chemical structure for molecule $x,$ stating an identity for the group $R.$

$($b$)$ Calculate the Index of Hydrogen Deficiency $($IHD$)$ for molecule $x$ and explain why it is consistent with the structure you have suggested in part $($a$).$

$($c$)$ All three molecules give rise to signals in the mass spectra at $\frac{m}{z}=120$ and $\frac{m}{z}=92.$ For each of the three molecules, identify the structures of the cations responsible for these two signals. Identify the fragmentation pathway responsible for the $\frac{m}{z}=120$ cation.