
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 angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $\frac{a}{\cos A}=\frac{b+c}{\cos B+\cos C}$.  
$(Ⅰ)$ Find the value of angle $A$.  
$(Ⅱ)$ Given $a=3$,  
$(i)$ If the area of $\triangle ABC$ is $\frac{\sqrt{3}}{2}$, find the perimeter of $\triangle ABC.  
$(ii)$ Find the range of possible values for the perimeter of $\triangle ABC.

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