
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
Consider the following propositions about the function $f(x) = 4\sin \left(2x+ \dfrac{\pi}{3}\right)$ for $x \in \mathbb{R}$, identify the correct ones:  
$(1)$ The expression of $y=f(x)$ can be rewritten as $y=4\cos \left(2x+ \dfrac{\pi}{3}\right)$ for $x \in \mathbb{R}$.  
$(2)$ The graph of $y=f(x)$ is symmetric about the point $\left(- \dfrac{\pi}{6},0\right)$.  
$(3)$ The minimal positive period of $y=f(x)$ is $2\pi$.  
$(4)$ One of the axes of symmetry for the graph of $y=f(x)$ is $x=- \dfrac{\pi}{6}$.

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