Trace minimization with additional constraint on $X$

Question

Consider the following SDP problem: \begin{align*} \min_X \; & \mathrm{Tr}[AX]\\ \mathrm{s.t.}\; & X \succeq 0\\ & X \succeq \begin{bmatrix}0 & 0.5\\0.5 & 0\end{bmatrix}, \end{align*} where $A$ is positive semidefinite. I know that the minimum is equal to $0$ if the only condition is the first constraint. Numerically, I know that there is a closed-form solution to this problem, which is \begin{equation*} 0.5(\sqrt{A_{11}} + A_{12}), \end{equation*} but I can't derive it from the above problem.