
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 $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. Let vector $m=(\cos (A-B),\sin (A-B))$ and $n=(\cos B,-\sin B)$, and it is given that $m\cdot n=-\frac{3}{5}$.  
$(1)$ Find the value of $\sin A$;  
$(2)$ If $a=4\sqrt{2}$ and $b=5$, find the magnitude of angle $B$ and the projection of vector $\overrightarrow{BA}$ in the direction of $\overrightarrow{BC}$.

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