
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]


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Given the function $f\left( x \right)=\sqrt{3}\sin x\cos x-\cos^{2}x-\dfrac{1}{2}$.

(1) Find the equation of the axis of symmetry for the function $f\left( x \right)$.
(2) Stretch the horizontal coordinates of the graph of $f\left( x \right)$ by a factor of 2 and then shift it left by $\dfrac{\pi }{3}$ units to obtain the graph of $g\left( x \right)$. If $a$, $b$, and $c$ are the lengths of the sides opposite to angles $A$, $B$, and $C$ in $\triangle ABC$, respectively, with $a=2$, $c=4$, and $g\left( B \right)=0$, find the value of $b$.

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