
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]


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Given the ellipse $ C $: $\frac {x^{2}}{a^{2}} + \frac {y^{2}}{b^{2}} = 1$ ($a > b > 0$) passing through the point $(1, \frac {3}{2})$, with eccentricity $e= \frac {1}{2}$.  
(Ⅰ) Find the equation of ellipse $C$;  
(Ⅱ) Assume a moving line $l: y=kx+m$ is tangent to the ellipse $C$ at point $T$, and the line $l$ intersects the line $x=4$ at point $S$. Is there a fixed point in the coordinate plane such that the circle with diameter $ST$ always passes through this fixed point? If it exists, find the coordinates of the point; if not, explain why.

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