
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 one focus of the ellipse ${C}_{1}: \frac{{x}^{2}}{{a}^{2}}+ \frac{{y}^{2}}{{b}^{2}}=1(a > b > 0) $ coincides with the focus of the parabola ${C}_{2}:{y}^{2}=2px(p > 0) $, and the distance from point $F$ to the line $x-y+1=0$ is $ \sqrt{2} $, and the length of the common chord of ${C}_{1} $ and ${C}_{2} $ is $2 \sqrt{6} $ .

(I) Find the equation of the ellipse ${C}_{1} $ and the coordinates of point $F$ ;

(II) A line $l$ passes through point $F$ and intersects with ${C}_{1} $ at points $A$ and $B$, and with ${C}_{2} $ at points $C$ and $D$. Find the range of values for $ \frac{1}{|AB|}+ \frac{1}{|CD|} $.

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