
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
Given a square $ABCD$ with side length $a > 1$. Points $A'$, $B'$, $C'$, and $D'$ are placed on the sides $[AB]$, $[BC]$, $[CD]$, and $[DA]$ at a distance of 1 from the vertices $A$, $B$, $C$, and $D$, respectively. This creates a smaller square $A'B'C'D'$. This construction is repeated iteratively. What is the smallest value of $a$ for which it is possible to iterate this process 2007 times?

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