
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
Among the following propositions related to conic sections, the ones that are true are numbered ______. (List all the correct proposition numbers).
① If A and B are two fixed points, and $|PA|-|PB|=2$, then the locus of point P is a hyperbola;
② If A and B are two fixed points, and $|PA|=10-|PB|$ with $|AB|=6$, then the maximum value of $|PA|$ is 8;
③ The roots of the equation $2x^2-5x+2=0$ can serve as the eccentricities of an ellipse and a hyperbola, respectively;
④ The hyperbola $\frac{x^2}{25}-\frac{y^2}{9}=1$ and the ellipse $x^2+\frac{y^2}{35}=1$ have the same foci.

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