Arithmetic Operation

Let $z_1=a+bi$ and $z_2=c+di$ be complex numbers, where $a$ , $b$ , $c$ and $d$ are real numbers and $a \neq 0$ and $c \neq 0$ . An arithmetic operation $\ast$ is defined as follows: $z_1 \ast z_2 = ac+(ad+bc)i.$ Let $z_3 = \frac{1}{2}-\frac{1}{4}i$ . If the inverse element of $z_3$ for $\ast$ is $p + qi$ , where $p$ and $q$ are real numbers, what is the value of $p^2 + q^2$ ?