
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 $C$: $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1\left(a \gt b \gt 0\right)$ with a focal length of $2$, and passing through the point $P(1$,$\frac{3}{2})$.<br/>$(1)$ Find the equation of the ellipse $C$;<br/>$(2)$ A moving line $l$ passing through the right focus $F$ of the ellipse with a slope of $k\left(k\neq 0\right)$ intersects the ellipse at points $A$ and $B$. Is there a fixed point $T$ on the $x$-axis, other than point $F$, such that $|AF|\cdot |BT|=|BF|\cdot |AT|$ always holds true? If yes, find the coordinates of point $T$; if not, explain the reason.

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