
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 slope of line \( l \) is \(\frac{1}{2}\) and it passes through the left focus of the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) (where \(a > b > 0\)), intersecting the ellipse at points \(A\) and \(B\). The center of the ellipse is \(O\), the distance from point \(O\) to the line \(AB\) is \( d = 1 \), and the length of chord \(AB\) is \(\frac{4}{5}\) of the major axis of the ellipse. Find the equation of the ellipse.

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