
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
Let $f(x)=\sin x \cdot \sin\left(\frac{\pi}{2}+x\right)+\sqrt{3}\cos^2 x-\frac{\sqrt{3}}{2}$.  
$(Ⅰ)$ Find the period and axis of symmetry of the graph of the function $f(x)$.  
$(Ⅱ)$ In acute triangle $\triangle ABC$, if $f(A)=0$ and the smallest circle that can cover $\triangle ABC$ has an area of $\pi$, find the range of $AB+AC$.

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