
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
16. The domain of function $f(x)$ is $D$. If there exists a closed interval $[a,b] \subseteq D$ such that the function $f(x)$ satisfies: (1) $f(x)$ is a monotonic function on $[a,b]$; (2) the range of $f(x)$ on $[a,b]$ is $[2a,2b]$, then the interval $[a,b]$ is called a "harmonious interval" of $y=f(x)$. Among the following functions, the ones that have a "harmonious interval" are __________. (Fill in the serial number only)  
(1) $f(x)=x^2$ $(x \geqslant 0)$; (2) $f(x)=e^x$ $(x \in \mathbb{R})$;  

(3) $f(x)=\dfrac{4x}{x^2+1}$ $(x \geqslant 0)$; (4) $f(x)=\log_a \left( a^x- \dfrac{1}{8} \right)$ $(a > 0$ and $a \neq 1)$.

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