
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 $\Delta ABC$, it is known that $\overrightarrow{AB} \cdot \overrightarrow{BC} = \frac{3}{2}S_{\Delta ABC}$, where $S_{\Delta ABC}$ is the area of $\Delta ABC$.

$(1)$ Find the value of $\sin B$;

$(2)$ If $A - C = \frac{\pi}{4}$, find the value of $\cos A$.

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