
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 foci of an ellipse are on the $y$-axis, the center is at the origin, the minor axis length is $2\sqrt{3}$, and the eccentricity $e= \frac{1}{2}$.

(I) Find the equation of the ellipse;

(II) If the line $l$ passing through point $P(0,3)$ intersects the ellipse at points $A$ and $B$, and point $M(2,0)$ satisfies $\overrightarrow{MA}+\overrightarrow{MB}=2\overrightarrow{MN}$, find the range of values for the $y$-intercept $d$ of the line $MN$.

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