
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  $ S$  be the set of points in the Cartesian plane that satisfy
\[ \Big|\big|{|x| \minus{} 2}\big| \minus{} 1\Big| \plus{} \Big|\big|{|y| \minus{} 2}\big| \minus{} 1\Big| \equal{} 1.
\]
If a model of  $ S$  were built from wire of negligible thickness, then the total length of wire required would be  $ a\sqrt {b},$  where  $ a$  and  $ b$  are positive integers and  $ b$  is not divisible by the square of any prime number.  Find  $ a \plus{} b.$ 

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