
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 a line segment $AB$ with length $3\sqrt{2}$, where the endpoints $A$ and $B$ move along the $x$-axis and $y$-axis respectively, and a moving point $P$ satisfies $\overrightarrow{BP} = 2\overrightarrow{PA}$. Let the trajectory of point $P$ be curve $C$.
$(1)$ Find the equation of curve $C$;
$(2)$ A line $l$ passing through point $(4,0)$ with a non-zero slope intersects curve $C$ at points $M$ and $N$. Is there a fixed point $T$ on the $x$-axis such that the product of the slopes of lines $MT$ and $NT$ is a constant? If it exists, find the coordinates of point $T$ and the constant; if not, explain why.

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