
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 $\triangle ABC$ have internal angles $A, B, C$ opposite to sides $a, b, c$ respectively, with $b=\sqrt{3}$.

$(1)$ If $C=\frac{5\pi}{6}$, and the area of $\triangle ABC$ is $\frac{\sqrt{3}}{2}$, find the length of side $c$.

$(2)$ If $B=\frac{\pi}{3}$, find the range of values for $2c-a$.

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