
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: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$) with an eccentricity of $\frac{\sqrt{2}}{2}$, where the left focus is F(-2, 0).  
(1) Find the equation of ellipse C;  
(2) If the line $y = x + m$ intersects ellipse C at two distinct points A and B, and the midpoint M of line segment AB lies on the circle $x^2 + y^2 = 1$, find the value of $m$.

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