
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 $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ has an eccentricity of $\frac{\sqrt{2}}{2}$, and the distance from one endpoint of the minor axis to a focus is $\sqrt{2}$.

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

(Ⅱ) Determine whether there exists a line $m$ that passes through the left focus of the ellipse $C$ and does not coincide with the $x$-axis, intersects the ellipse $C$ at points $M$ and $N$, the perpendicular bisector of segment $MN$ intersects the $x$-axis at point $P$ and intersects the ellipse $C$ at point $Q$, such that the quadrilateral $MPNQ$ is a rhombus. If such a line exists, find the equation of line $m$; if not, explain why.

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