
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
Equilateral triangle  $ABC$  has side length  $6$ . Circles with centers at  $A$ ,  $B$ , and  $C$  are drawn such that their respective radii  $r_A$ ,  $r_B$ , and  $r_C$  form an arithmetic sequence with  $r_A<r_B<r_C$ . If the shortest distance between circles  $A$  and  $B$  is  $3.5$ , and the shortest distance between circles  $A$  and  $C$  is  $3$ , then what is the area of the shaded region? Express your answer in terms of pi.

[asy]
size(8cm);
draw((0,0)--(6,0)--6*dir(60)--cycle);
draw(circle((0,0),1));
draw(circle(6*dir(60),1.5));
draw(circle((6,0),2));
filldraw((0,0)--arc((0,0),1,0,60)--cycle, grey);
filldraw(6*dir(60)--arc(6*dir(60),1.5,240,300)--cycle, grey);
filldraw((6,0)--arc((6,0),2,120,180)--cycle, grey);
label(" $A$ ",(0,0),SW);
label(" $B$ ",6*dir(60),N);
label(" $C$ ",(6,0),SE);
[/asy]

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