
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 $\overrightarrow{a}=({\cos\frac{{3x}}{2},\sin\frac{{3x}}{2}})$, $\overrightarrow{b}=({\cos\frac{x}{2},-\sin\frac{x}{2}})$, and $x∈[{0,\frac{π}{2}}]$.<br/>$(1)$ When $\overrightarrow{a}⊥\overrightarrow{b}$, find the value of $x$;<br/>$(2)$ When $|\overrightarrow{a}+\overrightarrow{b}|≥1$, find the range of $x$;<br/>$(3)$ If $f(x)=\overrightarrow{a}•\overrightarrow{b}-2m|\overrightarrow{a}+\overrightarrow{b}|$, $x∈[{0,\frac{π}{2}}]$, and the minimum value of $f\left(x\right)$ is $-2$, find the value of the real number $m$.

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