
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 $f(x)=\sin x \cos x - \sqrt{3} \cos^2 x$, when the graph of $f(x)$ is shifted to the left by $\frac{\pi}{4}$ units, the graph of the function $g(x)$ is obtained.  
$(1)$ Find the expression of the function $g(x)$ and its interval of monotonicity.  
$(2)$ For $x \in \left[\frac{\pi}{6}, \frac{\pi}{3}\right]$, if $af(x) + g(x) \geq \frac{\sqrt{a^2+1}}{2} - \frac{\sqrt{3}}{2}(a+1)$ always holds, find the range of positive $a$.

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