
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 $\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively. Given that $\frac{b}{{\sin A + \sin C}} = \frac{{a-c}}{{\sin B - \sin C}}$. Find:<br/>
$(1)$ The measure of angle $A$;<br/>
$(2)$ Let the area of $\triangle ABC$ be $S$. If $\overrightarrow{BM} = \frac{1}{2}\overrightarrow{MC}$, find the minimum value of $\frac{{{{|\overrightarrow{AM}|}^2}}}{S}$.

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