
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
Let vector $\overrightarrow{a}=(\cos (\alpha +\beta ),\sin (\alpha +\beta ))$, $\overrightarrow{b}=(\cos (\alpha -\beta ),\sin (\alpha -\beta ))$, and $\overrightarrow{a}+\overrightarrow{b}=(\frac{4}{5},\frac{3}{5})$.

$(1)$ Find $\tan \alpha$;

$(2)$ Find $\frac{2\cos^2\alpha -4\sin \alpha -1}{\sqrt{2}\sin (\alpha -\frac{\pi }{4})}$

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