
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
Let ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$), passing through point $Q(\sqrt{2},1)$, with the right focus $F(\sqrt{2},0)$,

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

(2) Let line $l$: $y = k(x-1)$ ($k > 0$) intersect the $x$-axis and $y$-axis at points $C$ and $D$ respectively, and intersect ellipse $C$ at points $M$ and $N$. If $\overrightarrow{CN} = \overrightarrow{MD}$, find the value of $k$ and the length of chord $|MN|$.

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