
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 the ellipse $\frac{x^{2}}{4}+ \frac{y^{2}}{3}=1$ with foci $F_1$ and $F_2$, and a parabola $C: x^2 = 2py$ with focus $F$ ($p>0$). $F_1F_2F$ forms an isosceles right triangle.

(I) Find the value of $p$.
(II) Consider a line $l$ passing through point $E(-2, 0)$ and intersecting parabola $C$ at points $A$ and $B$. Let $l_1$ and $l_2$ be the tangent lines to the parabola $C$ at points $A$ and $B$, respectively, such that $l_1 \perp l_2$. Determine whether such a line $l$ exists. If it exists, find its equation; otherwise, explain why not.

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