
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 $C: \frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with one of its foci coinciding with the focus of the parabola $y^{2}=8x$, and point $P(2, \sqrt{2})$ lies on $C$.

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

(2) If a chord of the ellipse $C$ is bisected at point $M(2,1)$, find the equation of the line containing this chord.

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