
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 $\overrightarrow{a}=(\cos x, \sqrt{2}\sin x)$ and $\overrightarrow{b}=(1, -\sqrt{2})$, where $x\in \left[0,\pi \right]$.<br/>$(1)$ If $(\overrightarrow{a}-\overrightarrow{b})∥\overrightarrow{b}$, find the value of the real number $x$;<br/>$(2)$ Given $\overrightarrow{c}=(m, -1)$ and $\overrightarrow{c}⊥\overrightarrow{b}$, if $f(x)=\overrightarrow{a}•\overrightarrow{c}$, find the range of $f\left(x\right)$.

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