
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 $f(x)=\sin x + \sqrt{3}\cos x$.  
$(1)$ Find the extreme values and intervals of monotonicity of $f(x)$.  
$(2)$ First, shift the graph of $y=f(x)$ to the left by $\frac{\pi}{3}$ units, then stretch all the horizontal coordinates of the points on the graph by a factor of $2$ (vertical coordinates remain unchanged). This results in the graph of the function $y=g(x)$. Find the value of $g\left(\frac{\pi}{6}\right)$.

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