<p>Find the largest possible real part of</p> <p><img alt="\[(75+117i)z+\frac{96+144i}{z}\]" height="37" src="https://latex.artofproblemsolving.com/8/9/f/89f4e07cc86e94772fd84e963c46fad30e9b51f7.png" width="194" /></p> <p>where <img alt="$z$" height="10" src="https://latex.artofproblemsolving.com/b/1/3/b13f21416d84e13708696f34dea81026cda583c9.png" width="11" /> is a complex number with <img alt="$|z|=4$" height="20" src="https://latex.artofproblemsolving.com/8/4/e/84ef5e8288cf9d83ed345ac49e48fb8a3e82cce5.png" width="55" />.</p>