
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 $A(-2,0)$, $B(2,0)$ in the plane, a moving point $P$ satisfies ${k_{PA}} \cdot {k_{PB}} = -\frac{3}{4}$.  
$(1)$ Find the trajectory equation of the moving point $P$;  
$(2)$ Given that the line $l$ intersects the curve $C$ at points $M$ and $N$, when the coordinates of $P$ are $({1,\frac{3}{2}})$, the condition $k_{PM}+k_{PN}=0$ always holds. Investigate whether the slope of the line $l$ is a constant value. If it is a constant value, find the value; if it is not a constant value, explain the reason.

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