
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
In the plane rectangular coordinate system $xOy$, given $E(-2,0)$, $F(2,0)$, $A(-1,0)$, a moving point $P$ satisfies $|PE|-|PF|=2$, and the locus of point $P$ is denoted as $\Gamma$.  
$(1)$ Find the equation of the curve $\Gamma$;  
$(2)$ If point $Q$ is also on the curve $\Gamma$, and $\overrightarrow{FP}=-3\overrightarrow{FQ}$, find the area of $\triangle APQ$;  
$(3)$ Is there a constant $\lambda$ such that $\angle PFA=\lambda \angle PAE$ always holds for the moving point $P$? Please provide your conclusion and reasoning.

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