
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 $a\in\left[\frac{\pi}{2},\pi\right]$, $\overrightarrow{a}=(2,-1)$, $\overrightarrow{b}=(\cos a,\sin a)$, and $\overrightarrow{a}\parallel \overrightarrow{b}$.

$(1)$ Find the value of $\tan \left(\alpha +\frac{\pi }{4}\right)$;    

$(2)$ Find the value of $\cos \left(\frac{5\pi }{6}-2\alpha \right)$.

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