
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 in $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are respectively $a$, $b$, and $c$; and it is known that $2\sin^2A + 3\cos(B+C) = 0$,

$(1)$ Find the size of angle $A$;

$(2)$ If the area of $\triangle ABC$, $S = 5\sqrt{3}$, and $a = \sqrt{21}$, find the value of $\sin B + \sin C$.

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