
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 $F_1$: $(x+ \sqrt {3})^2+y^2=9$ and $F_2$: $(x- \sqrt {3})^2+y^2=1$, and an ellipse $C$: $\frac {x^{2}}{a^{2}}+\frac {y^{2}}{b^{2}}=1$ ($a>b>0$) with its foci at the centers of $F_1$ and $F_2$, respectively, and passing through the intersection points of the two circles.  
(1) Find the equation of the ellipse $C$;  
(2) On the line $x=2 \sqrt {3}$, there are two points $M$ and $N$ ($M$ 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 $|MN|$ is minimized, find the length of the segment $MQ$.

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