
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
Determine the correct serial number(s) in the following propositions:

($①$) If $f(x) = \ln(2x)$, then $f'(x) = \frac{1}{x}$;

($②$) For all $a > 0$, the function $f(x) = (\ln x)^2 + \ln x - a$ has zero points;

($③$) If $f(x)$ is a differentiable function and its derivative $f'(x)$ is an even function, then the original function is an odd function;

($④$) $\int\_{-1}^{1}{\sqrt{4-{{x}^{2}}}}dx=\sqrt{3}+\frac{2\pi}{3}$.

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