
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 an ellipse $C$: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\left(a > b > 0\right)$ with left and right foci $F_{1}$ and $F_{2}$, eccentricity $\frac{\sqrt{2}}{2}$, a line $l$ passing through the left focus $F_{1}$ intersects the ellipse $C$ at points $A$ and $B$ ($A$, $B$ are not on the $x$-axis), and the perimeter of $\triangle ABF_{2}$ is $8\sqrt{2}$.<br/>$(1)$ Find the standard equation of the ellipse $C$;<br/>$(2)$ If point $P$ lies on the ellipse $C$ and $OP\bot AB(O$ is the origin), find the range of $\frac{|OP|^2}{|AB|}$.

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