
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 conclusion:
The moving point M(x, y) has the product of the slopes of the lines connecting it to the two fixed points (-4,0) and (4,0) equal to -$\frac{9}{16}$. Let the trajectory of M(x, y) be the curve C, and let F1 and F2 respectively be the left and right foci of the curve C. Among the propositions below:
(1) The coordinates of foci of curve C are F1(-5,0) and F2(5,0);
(2) There exists a point M on curve C such that the area of triangle $\triangle_{F1MF2} = 9$;
(3) P is a point on curve C, and P, F1, F2 form a right-angled triangle with |PF1| > |PF2|, and the value of $\frac{|PF1|}{|PF2|}$ is $\frac{23}{9}$;
(4) Let A(1,1), and the moving point P on curve C, the maximum value of |PA| + |PF1| is $8+\sqrt{9-2\sqrt{7}}$;
The correct propositions are numbered \_\_\_\_\_\_.

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