
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 with a perimeter of 24 is inscribed in a square with a perimeter of 32. The inner square is positioned such that its vertices touch the midpoints of the sides of the outer square. What is the greatest distance between a vertex of the inner square and a vertex of the outer square?
A) $7\sqrt{2}$
B) $8\sqrt{2}$
C) $5\sqrt{2}$
D) $9\sqrt{2}$
E) $6\sqrt{2}$

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