
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 an ellipse $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with an eccentricity of $\frac{1}{2}$. The line connecting one focus of the ellipse and the two vertices on the minor axis is tangent to the circle $x^{2}+y^{2}= \frac{3}{4}$.
(I) Find the equation of ellipse $C$.
(II) A line $l$ passing through point $(1,0)$ intersects $C$ at points $A$ and $B$. Is there a point $N$ on the x-axis such that $\overrightarrow{NA}\cdot \overrightarrow{NB}$ is a constant value? If so, find the coordinates of point $N$ and the constant value. If not, explain the reason.

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