
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 with its center at the origin $O$ and major axis length of $2\sqrt{2}$, with an eccentricity $e = \frac{\sqrt{2}}{2}$. A line $l$ passes through the right focus $F$ and intersects the ellipse at points $P$ and $Q$.
(1) Find the standard equation of the ellipse.
(2) When the slope of line $l$ is 1, find the chord length $|PQ|$.

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