
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
A square is called *proper* if its sides are parallel to the coordinate axes. Point  $P$  is randomly selected inside a proper square  $S$  with side length 2012. Denote by  $T$  the largest proper square that lies within  $S$  and has  $P$  on its perimeter, and denote by  $a$  the expected value of the side length of  $T$ . Compute  $\lfloor a \rfloor$ , the greatest integer less than or equal to  $a$ .

*Proposed by Lewis Chen*

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