
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
Determine the correct proposition numbers from the following:

1. The graph of an odd function always passes through the origin.
2. The function $y = \sqrt{x^2 - 1} + \sqrt{1 - x^2}$ is an even function, but not an odd function.
3. The graph of the function $f(x) = a^{x-1} + 3$ always passes through point $P$, and the coordinates of point $P$ are $(1, 4)$.
4. If $f(x + 1)$ is an even function, then $f(x + 1) = f(-x - 1)$.
5. If the function $f(x) = \begin{cases} a^x & (x > 1) \\ (4 - \frac{a}{2})x + 2 & (x \leq 1) \end{cases}$ is an increasing function on $\mathbb{R}$, then the range of the real number $a$ is $[4, 8)$.

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