
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. Given that $\cos^2 A - \cos^2 B - \cos^2 C = -1 + \sin B \sin C$. Find:

$(1)$ The value of angle $A$;

$(2)$ If $b + c = 6$, find the minimum value of the median $AM$ of triangle $\triangle ABC$.

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