
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 the ellipse $C:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1 (a > b > 0)$, its left and right foci are $F_1$ and $F_2$ respectively, and its top vertex is $A$. $\Delta AF_1F_2$ is an isosceles right triangle with a hypotenuse length of $2\sqrt{2}$.

$(1)$ Find the standard equation of the ellipse $C$;

$(2)$ If the line $l:y=x+m$ intersects the ellipse $C$ at two distinct points $P$ and $Q$.

    $(i)$ When $m=1$, find the length of the line segment $PQ$;

    $(ii)$ Is there a value of $m$ such that $S_{\Delta OPQ}=\dfrac{4}{3}$? If so, find the value of $m$; if not, explain why.

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