
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
A line with slope $k$ ($k \neq 0$) passes through the left focus $F$ of the ellipse $\frac{x^2}{2} + y^2 = 1$ and intersects the ellipse at points $A$ and $B$, such that the midpoint $M$ of $AB$ lies on the line $x + 2y = 0$. Find the value of $k$.

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