
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 $\triangle ABC$, it satisfies $\frac{{1+\sin A}}{{\cos A}}=\frac{{\sin 2B}}{{1-\cos 2B}}$.  
$(1)$ Find $\sin \left(C-B\right)$;  
$(2)$ Given point $D$ lies on side $BC$ and $AB:AD:AC=\sqrt{3}:\sqrt{2}:1$, find the value of $\frac{{CD}}{{BC}}$.

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