
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 that the center of the ellipse (C) is the origin (O), the major axis lies on the x-axis, the eccentricity $e= \frac{\sqrt{2}}{2}$, and the 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 the ellipse (C).
2. If the 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 values for the chord length $|GH|$ when the area of $\triangle{OEF}$ is maximum.

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