
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
Fido’s leash is now longer and allows him to reach the vertices of each side of his hexagonal yard. The yard remains a perfect hexagon. If the fraction of the area of Fido's yard that he is now able to explore while on his leash is expressed in simplest radical form as \(\frac{\sqrt{a}}{b}\pi\), what is the value of the product \(ab\)?

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