
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 about conic sections, the true propositions are numbered as ______. (Write down all the numbers of the true propositions).

① Given two fixed points $A$ and $B$, if $|PA|-|PB|=2$, then the trajectory of the moving point $R$ is a hyperbola;

② Given two fixed points $A$ and $B$, if the moving point $R$ satisfies $|PA|=10-|PB|$ and $|AB|=6$, then the maximum value of $|PA|$ is $8$;

③ The roots of the equation $2x^{2}-5x+2=0$ can be used as the eccentricities of an ellipse and a hyperbola, respectively;

④ The hyperbola $\dfrac{{x}^{2}}{25}- \dfrac{{y}^{2}}{9}=1$ and the ellipse ${{x}^{2}}+\dfrac{{{y}^{2}}}{35}=1$ have the same foci.

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