
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 lengths of the sides opposite to the internal angles $A$, $B$, $C$, respectively, and $2a\sin A = \left(2b+c\right)\sin B + \left(2c+b\right)\sin C$.  
$(1)$ Find the measure of angle $A$;  
$(2)$ If $a=7$ and the distance from vertex $A$ to side $BC$ is $\frac{15\sqrt{3}}{14}$, find the lengths of $b$ and $c$.

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