
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 point M(2, 0) and a circle C defined by $(x-a-1)^2+(y-\sqrt{3}a)^2=1$, there exists a point P on circle C such that the dot product $\overrightarrow{PM} \cdot \overrightarrow{PO}=8$ (O is the origin). Find the range of the real number $a$.

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