AMC 12 2025a Problem 24

A circle of radius $r$ is surrounded by $12$ circles of radius $1,$ externally tangent to the central circle and sequentially tangent to each other, as shown. Then $r$ can be written as $\sqrt a + \sqrt b + c,$ where $a, b, c$ are integers. What is $a+b+c?$

defaultpen(fontsize(12)+linewidth(1)); size(200);
real r=2.925, x=360/12;
pair O=origin;
draw(CR(O,r),black+1.5);
for (int i = 0; i<12; ++i) {
draw(CR((r+1)*dir(i*x),1));
}
dot(O); dot((r+1)*right);
draw(O--(r,0)^^(r+1,0)--(r+2,0), linewidth(0.5));
label("$r$",(r/2,0),up);
label("$1$",(r+3/2,0),up);

$\textbf{(A) } 3 \textbf{(B) } 5 \textbf{(C) } 7 \textbf{(D) } 9 \textbf{(E) } 11$