
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 $C$ with its center at the origin $O$, its foci on the x-axis, and its eccentricity is $\frac{1}{2}$. The distance from the left focus to the left vertex is 1.
(1) Find the standard equation of the ellipse $C$;
(2) A line passing through point $M(1, 1)$ intersects the ellipse $C$ at points $A$ and $B$, and point $M$ is the midpoint of chord $AB$. Find the equation of line $AB$.

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