
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 the following conclusions:  
A moving point $M(x,y)$ has the product of the slopes of the lines connecting it to two fixed points $(-4,0)$ and $(4,0)$ equal to $-\frac{9}{16}$. Let the trajectory of $M(x,y)$ be curve $C$, and let $F_1$ and $F_2$ respectively be the left and right foci of curve $C$. Among the following statements:  
$(1)$ The coordinates of the foci of curve $C$ are $F_1(-5,0)$, $F_2(5,0)$;  
$(2)$ There exists a point $M$ on curve $C$ such that the area of $\triangle_{F1MF2} = 9$;  
$(3)$ Let $P$ be a point on curve $C$, and $P, F_1, F_2$ are the vertices of a right triangle, with $|PF_1| > |PF_2|$, and the value of $\frac{|PF_1|}{|PF_2|}$ is $\frac{23}{9}$;  
$(4)$ Let $A(1,1)$, and let the moving point $P$ be on curve $C$, then the maximum value of $|PA| + |PF_1|$ is $8 + \sqrt{9 - 2\sqrt{7}}$;  
The correct statement numbers are ______.

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