
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 an ellipse $C: \frac{y^{2}}{a^{2}}+ \frac{x^{2}}{b^{2}}=1(a > b > 0)$ with an eccentricity of $\frac{\sqrt{2}}{2}$ and the sum of the distances from a point on the ellipse to the two foci is $2\sqrt{2}$. A line $l$ with slope $k(k\neq 0)$ passes through the upper focus of the ellipse and intersects the ellipse at points $P$ and $Q$. The perpendicular bisector of segment $PQ$ intersects the $y$-axis at point $M(0,m)$.

(1) Find the standard equation of the ellipse;

(2) Find the range of values for $m$;

(3) Express the area $S$ of triangle $\Delta MPQ$ in terms of $m$, and find the maximum value of the area $S$.

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