
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 $\triangle ABC$, $a=\sqrt{19}$, and $\frac{sinB+sinC}{cosB+cosA}=\frac{cosB-cosA}{sinC}$. Find:<br/>
$(1)$ The measure of angle $A$.<br/>
$(2)$ If point $D$ is a point on side $BC$ such that $\frac{BD}{DC}=\frac{3}{4}$ and $AD⊥AC$, find the area of triangle $\triangle ABC$.

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