
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 G: $$\frac{x^2}{2} + y^2 = 1$$, and a line \(l\) that does not coincide with the x-axis and passes through the left focus \(F_1\), intersecting the ellipse G at points A and B, the midpoint of chord AB is M, and the line OM intersects the ellipse G at points C and D.
1. If the slope of line \(l\) is 1, find the slope of line OM.
2. Is there a line \(l\) such that \(|AM|^2 = |CM| \cdot |DM|\)? If it exists, find the equation of line \(l\); if not, please explain why.

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