
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 $\frac{\sqrt{3}}{2}$, and the distance from one end of the minor axis to a focal point is 2.
(I) Find the equation of ellipse C.
(II) Given the line $l: x+4y-2=0$, a line $m$ is drawn through point $A(2,2)$ intersecting the ellipse $C$ at two distinct points $E$ and $F$, and intersecting line $l$ at point $K$. Determine whether there exists a constant $t$ such that $\frac{1}{|AE|} + \frac{1}{|AF|} = \frac{t}{|AK|}$ holds true, and explain the reason.

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