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

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