
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 $a=\left( \sin x, \frac{3}{4} \right)$, $b=(\cos x, -1)$.

$(1)$ When $a\parallel b$, find the value of $\tan \left( x- \frac{\pi}{4} \right)$;

$(2)$ Let the function $f(x)=2(a+b)\cdot b$, when $x\in \left[ 0, \frac{\pi}{2} \right]$, find the range of $f(x)$.

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