Each of the small circles in the figure has radius one. The innermost circle is tangent to the six circles that surround it, and each of those circles is tangent to the large circle and to its small-circle neighbors. Find the area of the shaded region.

[asy]
fill(Circle((0,0),3),gray(0.7));
fill(Circle((0,0),1),white);
fill(Circle((1.73,1),1),white);
fill(Circle((-1.73,1),1),white);
fill(Circle((-1.73,-1),1),white);
fill(Circle((1.73,-1),1),white);
fill(Circle((0,2),1),white);
fill(Circle((0,-2),1),white);
[/asy]
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
Each of the small circles in the figure has radius one. The innermost circle is tangent to the six circles that surround it, and each of those circles is tangent to the large circle and to its small-circle neighbors. Find the area of the shaded region.

[asy]
fill(Circle((0,0),3),gray(0.7));
fill(Circle((0,0),1),white);
fill(Circle((1.73,1),1),white);
fill(Circle((-1.73,1),1),white);
fill(Circle((-1.73,-1),1),white);
fill(Circle((1.73,-1),1),white);
fill(Circle((0,2),1),white);
fill(Circle((0,-2),1),white);
[/asy]
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
2\pi
Each of the small circles in the figure has radius one. The innermost circle is tangent to the six circles that surround it, and each of those circles is tangent to the large circle and to its small-circle neighbors. Find the area of the shaded region.

[asy]
fill(Circle((0,0),3),gray(0.7));
fill(Circle((0,0),1),white);
fill(Circle((1.73,1),1),white);
fill(Circle((-1.73,1),1),white);
fill(Circle((-1.73,-1),1),white);
fill(Circle((1.73,-1),1),white);
fill(Circle((0,2),1),white);
fill(Circle((0,-2),1),white);
[/asy]
train