
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 $E$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ that passes through the point $(-1, \frac{3}{2})$, and its eccentricity $e = \frac{1}{2}$.

(Ⅰ) Find the equation of the ellipse $E$;  
(Ⅱ) Let a variable line $l$: $y = kx + m$ be tangent to the ellipse $C$ at point $T$, and $l$ intersects the line $x = -4$ at point $S$.  
Is there a fixed point on the $x$-axis such that the circle with diameter $ST$ always passes through this fixed point? If it exists, find the coordinates of this point; if not, explain why.

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