
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 with its center at the origin and a focus on the x-axis, an eccentricity of $\sqrt{\frac{6}{3}}$, passing through the point $(\sqrt{5}, 0)$, and intersecting with the dynamic line through the fixed point $C(-1, 0)$ at points $A$ and $B$.

1. If the x-coordinate of the midpoint of segment $AB$ is $-\frac{1}{2}$, find the equation of line $AB$.
2. Does there exist a point $M$ on the x-axis such that $\overrightarrow{MA} \cdot \overrightarrow{MB}$ is constant? If so, find the coordinates of point $M$; if not, explain why.

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