
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
For functions $f(x)$ and $g(x)$ defined on the interval $D$, if for any $x \in D$, $|f(x)-g(x)| \leq 1$ holds, then the function $f(x)$ can be replaced by $g(x)$ on the interval $D$, which is called the "replacement interval". Consider the following propositions:  
① $f(x) = x^2 + 1$ on the interval $(-\infty, +\infty)$ can be replaced by $g(x) = x^2 + \frac{1}{2}$;  
② A "replacement interval" for $f(x) = x$ to be replaced by $g(x) = 1 - \frac{1}{4x}$ is $\left[ \frac{1}{4}, \frac{3}{2} \right]$;  
③ If $f(x) = \ln x$ on the interval $[1, e]$ can be replaced by $g(x) = \frac{1}{x} - b$, then $0 \leq b \leq \frac{1}{e}$;  
④ For $f(x) = \ln(ax^2 + x)$ ($x \in D_1$) and $g(x) = \sin x$ ($x \in D_2$), there exists a real number $a$ ($a \neq 0$) such that $f(x)$ can be replaced by $g(x)$ on the interval $D_1 \cap D_2$.  
Among these propositions, the true ones are $\boxed{\text{①②③}}$.

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