Find regular expressions for each of the following $($if exists$)$:

a$)$ Strings over the alphabet $\{x,y,z\}$ in which no $x\text{'}$s appear after the first $y.$

b$)$ Binary numbers that are multiples of $4.$

c$)$ Identifiers that start with a lowercase letter and then have $0$ or more alphanumeric characters, ending in a number.

d$)$ Strings over the alphabet $\{a,b\}$ that have exactly as many a$\text{'}$s as b$\text{'}$s