
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 the ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$) passes through the point $(2, \sqrt{2})$, and its eccentricity is $\frac{\sqrt{2}}{2}$.

(I) Find the equation of the ellipse $C$;

(II) Suppose a line passing through the left focus of the ellipse $C$ intersects the ellipse at points $M$ and $N$. The perpendicular bisector of segment $MN$ intersects the $y$-axis at point $P(0,m)$. Find the range of $m$.

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