
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
Given vectors $\overrightarrow{a}=(\cos x,\sin x)$ and $\overrightarrow{b}=(\sqrt{2}, \sqrt{2})$, if $\overrightarrow{a}\cdot \overrightarrow{b}= \frac{8}{5}$ and $\frac{\pi}{4} < x < \frac{\pi}{2}$,

$(1)$ try to find the values of $\cos (x- \frac{\pi}{4})$ and $\tan (x- \frac{\pi}{4})$;

$(2)$ find the value of $\frac{\sin 2x(1-\tan x)}{1+\tan x}$.

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