
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 \(\Omega\) be a circle of radius 8 centered at point \(O\), and let \(M\) be a point on \(\Omega\). Let \(S\) be the set of points \(P\) such that \(P\) is contained within \(\Omega\), or such that there exists some rectangle \(ABCD\) containing \(P\) whose center is on \(\Omega\) with \(AB = 4\), \(BC = 5\), and \(BC \parallel OM\). Find the area of \(S\).

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