
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 the ellipse \( C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \ (a > b > 0) \) has an eccentricity \( e = \frac{\sqrt{2}}{2} \). The line \( y = 2x - 1 \) intersects ellipse \( C \) at points \( A \) and \( B \), and the distance \( |AB| = \frac{8}{9} \sqrt{5} \).

(1) Find the equation of the ellipse \( C \);

(2) A line \( l \) passing through the point \( M(2,0) \) (with a non-zero slope) intersects the ellipse \( C \) at two distinct points \( E \) and \( F \) (with \( E \) between \( F \) and \( M \)). Let \( \lambda = \frac{S_{\triangle OME}}{S_{\triangle OMF}} \), determine the range of values for \( \lambda \).

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