
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 $\frac{1}{2}$, a circle with the origin as the center and the short semi-axis length of the ellipse as the radius, and a line $\sqrt{2}x - y + 3 = 0$ that is tangent to the circle.

(1) Find the equation of the ellipse $C$;
(2) Let a line passing through the point $(-1, 0)$ intersect the ellipse at points $A$ and $B$. Determine if there exists a fixed point $M$ on the $x$-axis such that $\overrightarrow{MA} \cdot \overrightarrow{MB}$ is constant. If it exists, find the constant value and the coordinates of point $M$. If not, explain the reason.

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