
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 $f(x)=\sqrt{3}\sin^2x+\sin x\cos x$.  
$(1)$ Find the interval where the function $f(x)$ is monotonically increasing.  
$(2)$ Stretch the abscissa of the points on the graph of the function $y=f(x)$ by a factor of $2$ (keeping the ordinate unchanged), then shift the resulting graph downward by $\frac{\sqrt{3}}{2}$ units to obtain the graph of the function $g(x)$. Find the minimum value of $g(x)$ and the set of values of $x$ at which this minimum value is attained.

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