
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
Given that the function $f(x)$ is defined on $\mathbb{R}$ and is not identically zero, and for any real numbers $x$, $y$, it satisfies: $f(2)=2$, $f(xy)=xf(y)+yf(x)$, $a_{n}= \dfrac {f(2^{n})}{2^{n}}(n\in\mathbb{N}^{*})$, $b_{n}= \dfrac {f(2^{n})}{n}(n\in\mathbb{N}^{*})$, consider the following statements:  
$(1)f(1)=1$; $(2)f(x)$ is an odd function; $(3)$ The sequence $\{a_{n}\}$ is an arithmetic sequence; $(4)$ The sequence $\{b_{n}\}$ is a geometric sequence.  
The correct statements are \_\_\_\_\_\_.

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