The intersection of the graphic solutions of all three inequalities listed below form an enclosed region in the coordinate plane.  What is the area of this region?  Express your answer in terms of $\pi$. \[
\left\{
\begin{array}{l}
(x-4)^2+y^2\leq 16 \\
y \geq x-4 \\
y\geq -\frac{1}{3}x
\end{array}
\right.
\]
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
The intersection of the graphic solutions of all three inequalities listed below form an enclosed region in the coordinate plane.  What is the area of this region?  Express your answer in terms of $\pi$. \[
\left\{
\begin{array}{l}
(x-4)^2+y^2\leq 16 \\
y \geq x-4 \\
y\geq -\frac{1}{3}x
\end{array}
\right.
\]
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
6\pi+2
The intersection of the graphic solutions of all three inequalities listed below form an enclosed region in the coordinate plane.  What is the area of this region?  Express your answer in terms of $\pi$. \[
\left\{
\begin{array}{l}
(x-4)^2+y^2\leq 16 \\
y \geq x-4 \\
y\geq -\frac{1}{3}x
\end{array}
\right.
\]
train