
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 a plane, there are $n$ circles, each two of which intersect at two points, and every three have no common point. They divide the plane into $f(n)$ regions, with $f(1)=2$, $f(2)=4$, $f(3)=8$. Find the expression for $f(n)$.

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