For a real number $a > \sqrt{2}$, define the function $f(x)$ as
$$f(x) = -x^3 + ax^2 + 2x$$
The tangent line to the curve $y = f(x)$ at the point $\mathrm{O}(0,0)$ intersects the curve $y = f(x)$ at another point A. The tangent line to the curve $y = f(x)$ at point A intersects the $x$-axis at point B. If point A lies on the circle with diameter OB, find the value of $\overline{\mathrm{OA}} \times \overline{\mathrm{AB}}$. [4 points]