
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
In triangle $\triangle ABC$, the sides opposite to $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $b^{2}=c\left(a+c\right)$.  
$(1)$ If $B=\frac{π}{3}$, find the value of $\frac{a}{c}$.  
$(2)$ If $\triangle ABC$ is an acute triangle, find the range of $\sqrt{3}\sin B+2\cos^2 C$.

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