
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: \\
(1) Let $A$ and $B$ be two fixed points, and $K$ be a non-zero constant. If $|PA| - |PB| = K$, then the trajectory of the moving point $P$ is a hyperbola; \\
(2) The roots of the equation $2x^{2} - 5x + 2 = 0$ can be used as the eccentricities of an ellipse and a hyperbola, respectively; \\
(3) The hyperbola $\frac{x^{2}}{25} - \frac{y^{2}}{9} = 1$ and the ellipse $\frac{x^{2}}{35} + y^{2} = 1$ have the same foci; \\
(4) Given the parabola $y^{2} = 2px$, if a circle is drawn with a chord $AB$ passing through the focus as its diameter, then this circle is tangent to the directrix. \\
The true propositions are __________ (Write down the numbers of all true propositions).

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