
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 \(X_{0}\) be the interior of a triangle with side lengths 3, 4, and 5. For all positive integers \(n\), define \(X_{n}\) to be the set of points within 1 unit of some point in \(X_{n-1}\). The area of the region outside \(X_{20}\) but inside \(X_{21}\) can be written as \(a\pi + b\), for integers \(a\) and \(b\). Compute \(100a + b\).

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