# Arithmetic Operation

Algebra Level 4

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 $$p^2 + q^2$$?

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