
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
Circles $A$, $B$, and $C$ are externally tangent to each other. Circle $A$ has a radius of 2. Circle $B$ has a radius of 1, and is externally tangent to an additional circle $E$, which also touches circle $A$ externally. All circles are in the same plane. Find the radius of circle $C$.

[asy]
unitsize(1cm);
pair A,B,C,E;
A=(0,0);
B=(3,0);
C=(1.5, 1.3228756555);
E=(-1,0);
draw(Circle(A,2),linewidth(0.7));
draw(Circle(B,1),linewidth(0.7));
draw(Circle(C,0.5),linewidth(0.7));
draw(Circle(E,1),linewidth(0.7));
label("$A$",A,W);
label("$B$",B,E);
label("$C$",C,N);
label("$E$",E,W);
[/asy]

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