For an ideal gas, consider only $P-V$ work in going from an initial state $X$ to the final state $Z$. The final state $Z$ can be reached by either of the two paths shown in the figure. Which of the following choice$(s)$ is (are) correct? [take $\Delta S$ as change in entropy and $w$ as work done].
$(A)$ $\Delta S_{X \to Z} = \Delta S_{X \to Y} + \Delta S_{Y \to Z}$
$(B)$ $w_{X \to Z} = w_{X \to Y} + w_{Y \to Z}$
$(C)$ $w_{X \to Y \to Z} = w_{X \to Y} + w_{Y \to Z}$
$(D)$ $\Delta S_{X \to Y \to Z} = \Delta S_{X \to Y}$

• A
  $(A, C)$
• B
  $(B, C)$
• C
  $(A, D)$
• D
  $(C, D)$