
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 $C$ with its center at the origin $O$, its major axis along the x-axis, and its eccentricity $e = \frac{\sqrt{2}}{2}$. A perpendicular line to the x-axis passing through the left focus $F_1$ intersects the ellipse at points $A$ and $A'$, with $|AA'| = \sqrt{2}$.
(1) Find the standard equation of ellipse $C$;
(2) If a line $l$ intersects the circle $O: x^2 + y^2 = 1$ at two distinct points $E$ and $F$, and intersects the ellipse $C$ at two distinct points $G$ and $H$, find the range of the chord length $|GH|$ when the area of $\triangle OEF$ is maximized.

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