
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 the ellipse $C$: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ with an eccentricity of $\frac{\sqrt{6}}{3}$ and a major axis length of $2\sqrt{3}$:

1. Find the equation of the ellipse.
2. If the line $y = kx - \frac{1}{2}$ intersects the ellipse $C$ at points $A$ and $B$, determine if there exists a fixed point $M$ on the positive half of the $y$-axis such that $\overrightarrow{MA} \perp \overrightarrow{MB}$. If it exists, find the coordinates of point $M$; if not, explain the reason.

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