
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 that the center of the ellipse $(C)$ is at the coordinate origin $O$, it shares the same foci with the hyperbola $({{x}^{2}}-8{{y}^{2}}=8)$, and passes through the point $(-2,\sqrt{2})$.

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

(2) A line $(l)$ passing through the left focus $({{F}\_{1}})$ of the ellipse $(C)$ intersects the ellipse at two points $P$ and $Q$. If a circle with $PQ$ as the diameter passes through the origin $O$, find the equation of the line $(l)$.

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