
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 $A$, $B$, $C$ are $a$, $b$, $c$ respectively. The vectors $m=(\cos (A-B),\sin (A-B))$, $n=(\cos B,-\sin B)$, and $m\cdot n=-\frac{3}{5}$.

(1) Find the value of $\sin A$.
(2) If $a=4\sqrt{2}$, $b=5$, find the measure of angle $B$ and the projection of vector $\overrightarrow{BA}$ onto the direction of $\overrightarrow{BC}$.

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