
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: $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ (where $a>b>0$), its left focus is F(-1, 0), O is the origin, and point $$G\left(1, \frac{\sqrt{2}}{2}\right)$$ lies on the ellipse. A line l passing through point F intersects the ellipse at two distinct points A and B.
(1) Find the equation of ellipse C.
(2) Find the equation of the trajectory of the midpoint M of chord AB.

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