
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 system $xOy$, there is an ellipse $E$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$) that passes through the point $(\sqrt{2}, 1)$ and is tangent to the line $\sqrt{2}x + 2y - 4 = 0$.

(1) Find the equation of ellipse $E$;  
(2) If ellipse $E$ intersects the x-axis at points $M$ and $N$, and there is a moving point $P$ inside ellipse $E$ such that $|PM|$, $|PO|$, and $|PN|$ form a geometric sequence, find the range of values for $\overrightarrow{PM} \cdot \overrightarrow{PN}$.

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