
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 opposite sides of angles $A$, $B$, and $C$ are $a$, $b$, $c$, $\overrightarrow{m}=(\sin A, \sin B)$, $\overrightarrow{n}=(\cos B, \cos A)$, $\overrightarrow{m} \cdot \overrightarrow{n} = -\sin 2C$.  
$(1)$ Find the value of angle $C$.  
$(2)$ Given $c = 2\sqrt{3}$ and $A = \frac{\pi}{4}$, find the value of $b$.

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