
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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