
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: $\frac{x^2}{2} + y^2 = 1$.
(I) Find the lengths of the major and minor axes, the eccentricity $e$, and the left focus $F_1$ of the ellipse C.
(II) Through the left focus $F_1$ of the ellipse C, construct line $l$. Line $l$ intersects the ellipse C at points A and B. If $|AB| = \frac{8\sqrt{2}}{7}$, find the equation of line $l$.

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