
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 circle $M: x^{2}+y^{2}-2 \sqrt {3}x=0$ is the right focus of the ellipse $C: \frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1 (a > b > 0)$, and the line passing through the left focus and the upper vertex of the ellipse is tangent to the circle $M$.
(I) Find the equation of the ellipse $C$;
(II) There are two points $A(x_{1},y_{1})$ and $B(x_{2},y_{2})$ on the ellipse $C$, and the product of the slopes of $OA$ and $OB$ is $-\frac{1}{4}$. Find the value of $x_{1}^{2} + x_{2}^{2}$.

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