
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
Spot's doghouse has a regular hexagonal base where each side measures two yards. He is tethered to a vertex with a four-yard rope. A bush is planted 1-yard from the vertex to which he is tethered in the opposite direction from the center of the base, effectively obstructing a small circular region. What is the area, in square yards, of the region outside the doghouse that Spot can reach? Assume the bush covers a $120^\circ$ circular sector with a 1-yard radius, from the point Spot is tethered. Express your answer in terms of $\pi$.

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