
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 to the internal angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and $S$ is the area of $\triangle ABC$. It is given that $b\cos C - c\cos B = 2a$.  
$(1)$ If $c = a$, find the value of angle $B$.  
$(2)$ If $c \leqslant 2a$, a perpendicular line is drawn from point $B$ to side $AB$ intersecting $AC$ at point $D$. Find the range of values for $\frac{BC \cdot BD}{S}$.

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