
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 that the distance from moving point $P$ to two fixed points $A({-2\sqrt{2},0})$ and $B({2\sqrt{2},0})$ is $6$, and the trajectory of moving point $P$ is a curve.  
$(1)$ Find the equation of curve $C$;  
$(2)$ Line $l:x-my-1=0$ intersects curve $C$ at points $M$ and $N$. Does there exist a point $T$ on the $x$-axis (if we denote the slopes of lines $MT$ and $NT$ as $k_{MT}$ and $k_{NT}$, respectively) such that $k_{MT}\cdot k_{NT}$ is a constant value? 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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