
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
Let $f(x)$ be a function defined on domain $D$. If there exists an interval $[m, n] \subset D$ ($m < n$) such that the range of the function $f(x)$ on $[m, n]$ is exactly $[km, kn]$, then the function $f(x)$ is called a $k$-type function.
1. $f(x) = 3 - \frac{4}{x}$ cannot be a $k$-type function;
2. If the function $y = -\frac{1}{2}x^2 + x$ is a $3$-type function, then $m = -4$, $n = 0$;
3. Suppose the function $f(x) = |3^x - 1|$ is a $2$-type function, then $m + n = 1$;
4. If the function $y = \frac{(a^2 + a)x - 1}{a^2x}$ ($a \neq 0$) is a $1$-type function, then the maximum value of $n - m$ is $\frac{2\sqrt{3}}{3}$.

The correct sequence numbers are __________.

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