A function \(f\) is defined by \(f(z) = (4 + i) z^2 + \alpha z + \gamma\) for all complex numbers \(z\), where \(\alpha\) and \(\gamma\) are complex numbers and \(i^2 = -1\). Suppose that \(f(1)\) and \(f(i)\) are both real. The original answer is in the form \(\sqrt{m}\). Find the smallest possible value of \(m\).

Let's think step by step and output the final answer within \boxed{}.
user