
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
Given circles $A, B,$ and $C$ have radii 1, 2, and 1.5 respectively, and circle $A$ and circle $B$ share a common point of tangency, and circle $C$ is tangent to $\overline{AB}$ at a point $N$ where $N$ is one-third the distance from $A$ to $B$, calculate the area inside circle $C$ but outside circle $A$ and circle $B$.

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