Complex Number Operations and Real Part Extraction

Consider the complex number defined as:

$$ z = 3 + 4i $$

where $i$ is the imaginary unit. You are tasked with computing the sum of $z$ and its conjugate $\overline{z}$, then multiplying the result by the complex number $1 - 2i$. Finally, find the real part of the final result and express it as a reduced fraction.