
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 $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and $A$, $B$, $C$ form an arithmetic sequence. The area of $\triangle ABC$ is $\frac{\sqrt{3}}{2}$.

$(1)$ Find the value of $ac$;

$(2)$ If $b=\sqrt{3}$, find the values of $a$ and $c$ $(a > c)$.

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