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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