
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 $\triangle ABC$, let $a$, $b$, and $c$ be the lengths of the sides opposite angles $A$, $B$, and $C$, respectively. Given vectors $\overrightarrow{m}=(\cos A, a)$ and $\overrightarrow{n}=(2\cos C-3\cos B, 3b-2c)$, and $\overrightarrow{m}$ is parallel to $\overrightarrow{n}$.  
$(1)$ Find $\frac{b}{c}$.  
$(2)$ If $A=\frac{2\pi}{3}$ and the area of $\triangle ABC$ is $\frac{2\sqrt{3}}{3}$, find the value of $a$.

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