
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 angles $A$, $B$, $C$ opposite to sides $a$, $b$, $c$ respectively, satisfy $2 \overrightarrow{AB} \cdot \overrightarrow{AC} = a^2 - (b+c)^2$.

$(1)$ Find the size of angle $A$;

$(2)$ Find the maximum value of $\sin A \cdot \sin B \cdot \sin C$, and the sizes of angles $B$, $C$ when the maximum value is achieved.

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