
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 $E:\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1(a>b>0)$ with a major axis length of $4$, and the point $P(1,\frac{3}{2})$ lies on the ellipse $E$. <br/>$(1)$ Find the equation of the ellipse $E$; <br/>$(2)$ A line $l$ passing through the right focus $F$ of the ellipse $E$ is drawn such that it does not coincide with the two coordinate axes. The line intersects $E$ at two distinct points $M$ and $N$. The perpendicular bisector of segment $MN$ intersects the $y$-axis at point $T$. Find the minimum value of $\frac{|MN|}{|OT|}$ (where $O$ is the origin) and determine the equation of line $l$ at this point.

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