
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 $x \in D$, the inequality $|f(x)-g(x)| \leqslant 1$ holds, then the function $f(x)$ can be replaced by $g(x)$ on the interval $D$, and $D$ is called the "replacement interval". Consider the following propositions:  
1. $f(x)=x^{2}+1$ can be replaced by $g(x)=x^{2}+ \frac {1}{2}$ on the interval $(-\infty,+\infty)$;  
2. A "replacement interval" for $f(x)=x$ to be replaced by $g(x)=1- \frac {1}{4x}$ is $[\frac {1}{4}, \frac {3}{2}]$;  
3. If $f(x)=\ln x$ can be replaced by $g(x)= \frac {1}{x}-b$ on the interval $[1,e]$, then $0 \leqslant b \leqslant \frac {1}{e}$;  
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 $a (\neq 0)$ 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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