
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 $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. Given that $c = \sqrt{2}$, $\cos C = \frac{3}{4}$, and $2c\sin A = b\sin C$. Find:
- $(1)$ the value of $b$;
- $(2)$ the value of $\sin A$;
- $(3)$ the value of $\sin\left(2A + \frac{\pi}{3}\right)$.

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