MathDB

As shown in the diagram, three circles of radius

1

are all externally tangent to each other, and there are two circles that are tangent to all three of these circles. Find the area of the shaded region.
[asy] size(5cm); real r=2/sqrt(3)-1, R=2/sqrt(3)+1; pair O=(0,0), C_1=O+(0,2/sqrt(3)), C_2=O+(-1,-1/sqrt(3)), C_3=O+(1,-1/sqrt(3)); fill(circle(O,R),rgb(0.5,0.5,0.5)); draw(circle(O,R)); fill(circle(C_1,1),rgb(1,1,1)); draw(circle(C_1,1)); fill(circle(C_2,1),rgb(1,1,1)); draw(circle(C_2,1)); fill(circle(C_3,1),rgb(1,1,1)); draw(circle(C_3,1)); fill(circle(O,r),rgb(1,1,1)); draw(circle(O,r)); [/asy]

Problem Statement