
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 $E$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ has an eccentricity of $\frac{\sqrt{3}}{2}$, and it passes through the point $(1, \frac{\sqrt{3}}{2})$.

(Ⅰ) Find the equation of $E$;  
(Ⅱ) Determine whether there exists a line $l$: $y=kx+m$ that intersects at points $P$ and $Q$, and satisfies: ① The sum of the slopes of $OP$ and $OQ$ ($O$ is the origin) is $2$; ② The line $l$ is tangent to the circle $x^2+y^2=1$. If it exists, find the equation of $l$; if not, explain why.

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