
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
Consider the following propositions:
1. Let A and B be two fixed points, and let k be a non-zero constant. If the difference between the distances from a moving point P to points A and B is k, i.e., $|\vec{PA}| - |\vec{PB}| = k$, then the locus of point P is a hyperbola.
2. If the sum of the first n terms of a geometric sequence is $s_n = 2^n + k$, then it must be the case that $k = -1$.
3. For all positive real numbers x, the minimum value of $2^x + 2^{-x}$ is 2.
4. 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.
5. The locus of points in the plane that are equidistant from a fixed point (3, -1) and a fixed line $x + 2y - 1 = 0$ is a parabola.

The correct propositions are denoted by their numbers _______.

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