
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
Let $\triangle ABC$ have sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$, respectively. Given that $\sqrt{b+c}$ is the geometric mean of $2a$ and $\sin\left(C+\frac{\pi}{6}\right)$, find:

$(1)$ The value of angle $A$;

$(2)$ If $\triangle ABC$ is an acute triangle and $c=2$, find the range of values for $a\sin B$.

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