
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 sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively, with $2a-c=2b\cos C$.  
$(1)$ Find angle $B$;  
$(2)$ If point $D$ is the midpoint of side $BC$, and points $E$ and $F$ lie on sides $AB$ and $AC$, respectively, such that $\angle EDF=\frac{\pi}{3}$ and $b=c=2$. Let $\angle BDE=\alpha$. Express the area $S$ of $\triangle DEF$ as a function of $\alpha$ and determine the range of values for $S$.

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