
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
Three circles with centers \( A, B \), and \( C \) have radii \( \sqrt{3}-1 \), \( 3-\sqrt{3} \), and \( 1+\sqrt{3} \), respectively. Each circle is externally tangent to the other two. The area of the shaded region is of the form \( a \sqrt{3} + b \pi + c \pi \sqrt{3} \) for some rational numbers \( a, b \), and \( c \). What is the value of \( a+b+c \)?

(Note: The provided image appears to be missing, but the problem can still be analyzed based on the given information.)

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