
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
(1) The solution set of the inequality $|2x-3| < 1$ is _______.

(2) Given $\tan \alpha=2$, and $\tan (\alpha +\beta )=\frac{1}{5}$, then the value of $\tan \beta$ is _______.

(3) The range of the function $f(x)=\sin x\cos x+\sin x+\cos x$ is _______.

(4) For functions $f(x)$ and $g(x)$ and interval $D$, if there exists $x_{0}\in D$ such that $|f(x_{0})-g(x_{0})|\leqslant 1$, then $x_{0}$ is called the "adjacent point" of the functions $f(x)$ and $g(x)$ on interval $D$. Given two functions:

① $f(x)=x^{2}-1$, $g(x)=2x-3$;

② $f(x)=x+1$, $g(x)=\sin x$;

③ $f(x)=e^{-x}$, $g(x)=-\frac{1}{x}$;

④ $f(x)=\ln x$, $g(x)=x$.

Then, on the interval $(0,+\infty)$, the ones that have a unique "adjacent point" are _______. (Fill in all correct serial numbers).

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