
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 four propositions:
1. The function $f(x) = a^x$ ($a>0$ and $a \neq 1$) shares the same domain as the function $g(x) = \log_a a^x$ ($a>0$ and $a \neq 1$).
2. The function $f(x) = x^3$ shares the same range as the function $g(x) = 3^x$.
3. Both functions $f(x) = (x-1)^2$ and $g(x) = 2^{x-1}$ are increasing on the interval $(0, +\infty)$.
4. If a function $f(x)$ has an inverse $f^{-1}(x)$, then the inverse of $f(x+1)$ is $f^{-1}(x+1)$.
Identify the incorrect proposition(s).

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