
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 two circles \\({{F}_{1}}:{{\left( x+\sqrt{3} \right)}^{2}}+{{y}^{2}}=9\\) and \\({{F}_{2}}:{{\left( x-\sqrt{3} \right)}^{2}}+{{y}^{2}}=1\\), and an ellipse \\(C:\dfrac{{x}^{2}}{{a}^{2}}+\dfrac{{y}^{2}}{{b}^{2}}=1 \left( a > b > 0 \right)\\) with the centers of circles \\({F}_{1},{F}_{2} \\) as its foci, which passes through the intersection points of the two circles.

(Ⅰ) Find the equation of the ellipse \\(C\\);

(Ⅱ) On the line \\(x=2\sqrt{3}\\), there are two points \\(M\\), \\(N\\) (\\(M\\) is in the first quadrant) such that \\(\overrightarrow{{F}_{1}M} \cdot \overrightarrow{{F}_{2}N}=0\\), and the line \\(MF_{1}\\) intersects with \\(NF_{2}\\) at point \\(Q\\). When \\(\left| MN \right|\\) is minimized, find the length of segment \\(MQ\\).

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