
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 moving circle M that is externally tangent to the circle O₁: $x^2 + y^2 + 6x + 5 = 0$ and internally tangent to the circle O₂: $x^2 + y^2 - 6x - 91 = 0$, the curve C represents the trajectory of the moving circle’s center M. Among the following propositions:
(1) The trajectory equation of the moving circle M is $\frac{x^2}{36} + \frac{y^2}{27} = 1$;
(2) If $\angle O_1MO_2 = 60°$, then the area of $\triangle O_1MO_2$ is $27 \sqrt{3}$;
(3) A circle with the origin as its center and a radius of 6 has no common points with curve C;
(4) For the moving point M $(x, y)$, where $y \neq 0$, the product of the slopes of the lines connecting M with the two fixed points $(-6, 0)$ and $(6, 0)$ is $-\frac{3}{4}$,
the correct proposition numbers are: ______.

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