
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
Given the function $f(x)=\lambda \sin \omega x+\cos \omega x$ $(\omega > 0)$, the distance between two adjacent axes of symmetry of its graph is $\dfrac{\pi }{2}$, and the line $x=\dfrac{\pi }{6}$ is one of its axes of symmetry.

$(1)$ Find the value of the real number $\lambda$;

$(2)$ Let the function $g(x)=f(x)+\cos (2x-\dfrac{\pi }{3})$, find the range of $g(x)$ in the interval $\left[-\dfrac{\pi }{2},\dfrac{\pi }{12}\right]$.

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