
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
In the Cartesian coordinate plane $xOy$, points $A$ and $B$ lie on the circle $x^2+y^2=4$, and $AB=2\sqrt{2}$. Point $P$ has coordinates $(3,-1)$, and $\overrightarrow{PO} \cdot (\overrightarrow{PA}+\overrightarrow{PB})=16$. If $M$ is the midpoint of $AB$ with an $x$-coordinate of $x_0$, find all possible values of $x_0$.

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