
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 a fixed point $C(-1,0)$ and an ellipse $x^{2}+3y^{2}=5$, a moving straight line passes through point $C$ and intersects the ellipse at points $A$ and $B$.
(I) If the x-coordinate of the midpoint of segment $AB$ is $-\frac{1}{2}$, find the equation of line $AB$.
(II) 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 the reason.

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