
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
The following are four 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 locus of the moving point $P$ is a hyperbola;
(2) The two roots of the equation $2x^2 - 5x + 2 = 0$ can respectively serve as the eccentricity of an ellipse and a hyperbola;
(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) The left and right foci of the ellipse $\frac{x^2}{16} + \frac{y^2}{7} = 1$ are respectively $F_1$ and $F_2$. The chord $AB$ passes through the point $F_1$, and if the length of the inscribed circle of $\triangle AB F_2$ is $\pi$, then the absolute difference between the y-coordinates of points $A$ and $B$ is $\frac{4}{3}$. Write down the serial numbers of all true propositions.

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