
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
Given that $a$, $b$, $c$ are the three sides of triangle $\triangle ABC$ opposite to the internal angles $A$, $B$, $C$ respectively, and $\frac{{a-c}}{{b+c}}=\frac{{\sin B}}{{\sin A+\sin C}}$.  
$(1)$ Find the value of angle $A$;  
$(2)$ Let $D$ be a point on side $BC$ such that $DA \perp BA$ and $BD=3DC$. Find the value of $\cos C$.

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