
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 $C$: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$, passing through point $Q(\sqrt{2}, 1)$ and having the right focus at $F(\sqrt{2}, 0)$,
(I) Find the equation of the ellipse $C$;
(II) Let line $l$: $y = k(x - 1) (k > 0)$ intersect the $x$-axis, $y$-axis, and ellipse $C$ at points $C$, $D$, $M$, and $N$, respectively. If $\overrightarrow{CN} = \overrightarrow{MD}$, find the value of $k$ and calculate the chord length $|MN|$.

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