
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 propositions, the correct serial number is ______.

(1) There exists $x_{0} > 0$ such that $x_{0} < \sin x_{0}$.

(2) If $\sin \alpha \neq \frac{1}{2}$, then $\alpha \neq \frac{\pi}{6}$.

(3) "$\ln a > \ln b$" is the necessary and sufficient condition for "$10^{a} > 10^{b}$".

(4) If the function $f(x)=x^{3}+3ax^{2}+bx+a^{2}$ has an extreme value of $0$ at $x=-1$, then $a=2$, $b=9$ or $a=1$, $b=3$.

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