
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 $k$ circles, and every two circles intersect at two points, with no three circles passing through the same point. If the $k$ circles divide the plane into $f(k)$ regions, then $k+1$ circles will divide the plane into $f(k) + \_\_\_\_\_\_$ regions.

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