
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
In the diagram below, the circle with center  $A$  is congruent to and tangent to the circle with center  $B$ .  A third circle is tangent to the circle with center  $A$  at point  $C$  and passes through point  $B$ .  Points  $C$ ,  $A$ , and  $B$  are collinear.  The line segment  $\overline{CDEFG}$  intersects the circles at the indicated points.  Suppose that  $DE = 6$  and  $FG = 9$ .  Find  $AG$ .
[asy]
unitsize(5);
pair A = (-9 sqrt(3), 0);
pair B = (9 sqrt(3), 0);
pair C = (-18 sqrt(3), 0);
pair D = (-4 sqrt(3) / 3, 10 sqrt(6) / 3);
pair E = (2 sqrt(3), 4 sqrt(6));
pair F = (7 sqrt(3), 5 sqrt(6));
pair G = (12 sqrt(3), 6 sqrt(6));
real r = 9sqrt(3);
draw(circle(A, r));
draw(circle(B, r));
draw(circle((B + C) / 2, 3r / 2));
draw(C -- D);
draw(" $6$ ", E -- D);
draw(E -- F);
draw(" $9$ ", F -- G);
dot(A);
dot(B);
label(" $A$ ", A, plain.E);
label(" $B$ ", B, plain.E);
label(" $C$ ", C, W);
label(" $D$ ", D, dir(160));
label(" $E$ ", E, S);
label(" $F$ ", F, SSW);
label(" $G$ ", G, N);
[/asy]

Present the answer in LaTex format: \boxed{Your answer}
user