
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 the complex number corresponding to the vector $\overrightarrow{OP}$ in the complex plane be $z_{P}$. Given a nonzero real number $z$, the vector $z(\overrightarrow{OP})=({{{Re}}({z•{z_P}}),{{Im}}({z•{z_P}})})$ is called the $z$-vector of $\overrightarrow{OP}$.
$(1)$ Given $\overrightarrow{OA}=({{x_0},{y_0}})$, find $z(\overrightarrow{OA})$;
$(2)$ Let $\overrightarrow{v}=(x,y)(x>0,y>0),\overrightarrow{i}=(1,0),\overrightarrow{j}=(0,1)$ have $z$-vectors $\overrightarrow{OV'},\overrightarrow{OE},\overrightarrow{OF}$ respectively, with $S_{\triangle OV'E}=1$ and $S_{\triangle OV'F}=2$. Find the coordinates of $\overrightarrow{v}$ (express the result in terms of $z$);
$(3)$ If for all $A,B$ such that $S_{\triangle OAB}=1$, the minimum value that can be achieved for $z(\overrightarrow{OA})•\overrightarrow{OA}+z(\overrightarrow{OB})•\overrightarrow{OB}$ is $8$, find the value of the real number $z$.

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