
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 $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and angle $A$ satisfies: $2\cos^2A-2\sqrt{3}\sin A\cos A=-1$.
$(1)$ If $a=2\sqrt{3}$ and $c=2$, find the area of $\triangle ABC$;
$(2)$ Find the value of $\dfrac{b-2c}{a\cdot\cos \left(60^{\circ}+C\right)}$.

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