Complex problem in complex numbers

Algebra Level 5

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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