
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
Given a triangle $ABC$ with its three internal angles $A$, $B$, $C$ and the opposite sides $a$, $b$, $c$ respectively, and the area of $\triangle ABC$ is $S= \frac { \sqrt {3}}{2}ac\cos B$.
$(1)$ If $c=2a$, find the sizes of angles $A$, $B$, $C$;
$(2)$ If $a=2$, and $\frac {\pi}{4}\leqslant A\leqslant \frac {\pi}{3}$, find the range of values for side $c$.

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