
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 $\overrightarrow{a}=(\sin x,1)$ and $\overrightarrow{b}=(\cos x,2)$, where $x\in \mathbb{R}$, and the function $f(x)=\overrightarrow{a}\cdot \overrightarrow{b}$.

$(1)$ When $x\in \left[ -\frac{\pi}{12},\frac{\pi}{3} \right]$, find the maximum and minimum values of $|\overrightarrow{a}+\overrightarrow{b}|$;

$(2)$ If $f(\alpha)=\frac{12}{5}$, where $\alpha \in \left( \frac{\pi}{4},\frac{\pi}{2} \right)$, find $\tan \left( 2\alpha +\frac{3\pi}{4} \right)$

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