
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 $C: \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ $(a > b > 0)$ with its left and right foci as $F_{1}$ and $F_{2}$ respectively, and its top vertex as $A$. The triangle $\triangle AF_{1}F_{2}$ is an isosceles right triangle with the hypotenuse of length $2\sqrt{2}$.
   (I) Find the standard equation of the ellipse $C$;
   (II) Let a line $l: y = x + m$ intersect the ellipse $C$ at two distinct points $P$ and $Q$.
   (i) When $m = 1$, find the length of segment $PQ$;
   (ii) Determine whether there exists a value for $m$ such that the area of $\triangle OPQ$ is $\frac{4}{3}$. If such an $m$ exists, find its value; if not, provide a justification.

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