The functions $u$ and $w$ are defined as follows. \[ \begin{array}{l} u(x)=-x-1 \\ w(x)=x^{2}+2 \end{array} \] Find the value of $u(w(3))$.

Step 1 ：Define the functions \(u(x) = -x - 1\) and \(w(x) = x^2 + 2\)

Step 2 ：Find the value of \(w(3)\) by substituting \(x = 3\) into the function \(w(x)\)

Step 3 ：Substitute the value of \(w(3)\) into the function \(u(x)\) to find the value of \(u(w(3))\)

Step 4 ：Calculate the final value of \(u(w(3))\)

Step 5 ：\(\boxed{-12}\) is the final answer