
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
Let \(\Gamma\) be a circle of radius 1 centered at \(O\). A circle \(\Omega\) is said to be friendly if there exist distinct circles \(\omega_{1}, \omega_{2}, \ldots, \omega_{2020}\), such that for all \(1 \leq i \leq 2020\), \(\omega_{i}\) is tangent to \(\Gamma\), \(\Omega\), and \(\omega_{i+1}\). (Here, \(\omega_{2021}=\omega_{1}\)). For each point \(P\) in the plane, let \(f(P)\) denote the sum of the areas of all friendly circles centered at \(P\). If \(A\) and \(B\) are points such that \(O A=\frac{1}{2}\) and \(O B=\frac{1}{3}\), determine \(f(A)-f(B)\).

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