Complex problem in complex numbers

Algebra Level 4

For \(i = \sqrt{-1} \), let \(z\) be a complex number satisfying \(z = a^i \). Denote \(f(x) \) and \(g(x) \) as the real part and imaginary part of \(z\) respectively, that is \(f(x) = \Re(z) , g(x) = \Im(z) \).

Let the intersection point of \(f(x) \) and \(g(x) \) be expressed as \((m,n) \) in the coordinate axes.

Denote \(m_n \) the \(n^\text{th} \) absissa, where \(m_n < 1 \). Compute \( m_1 + m_2 + m_3 + \cdots \).

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