
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
Among the following statements:

1. If $f(x)=ax^{2}+(2a+b)x+2$ (where $x\in[2a-1,a+4]$) is an even function, then the real number $b=2$;
2. $f(x)= \sqrt {2008-x^{2}}+ \sqrt {x^{2}-2008}$ is both an odd function and an even function;
3. Given that $f(x)$ is defined on $\mathbb{R}$ as an odd function, if $f(x)=x(1+x)$ when $x\in[0,+\infty)$, then $f(x)=x(1+|x|)$ when $x\in\mathbb{R}$;
4. Given that $f(x)$ is defined on $\mathbb{R}$ as a non-zero constant function, and for any $x$, $y\in\mathbb{R}$, it satisfies $f(x\cdot y)=x\cdot f(y)+y\cdot f(x)$, then $f(x)$ is an odd function.

Determine the serial number(s) of the correct statement(s). (Note: Fill in the serial number(s) you believe to be correct)

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