
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 lengths of the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, $c$, and $\overrightarrow{BA} \cdot \overrightarrow{BC} = \frac{3}{2}$, and the area of $\triangle ABC$ is $\frac{3\sqrt{3}}{4}$.  
$(1)$ If $b = \sqrt{3}$, find the value of $a+c$;  
$(2)$ Find the range of values for $2\sin A - \sin C$.

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