
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
Let the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \left( a > b > 0 \right)$ pass through the point $\left( 1, \frac{\sqrt{6}}{2} \right)$, and its eccentricity equals $\frac{\sqrt{2}}{2}$.

$(1)$ Find the equation of the ellipse $C$;

$(2)$ Through point $P(2,0)$, draw lines $PA$ and $PB$ intersecting the ellipse at points $A$ and $B$, respectively, and satisfying $PA \perp PB$. Determine whether the line $AB$ passes through a fixed point. If it does, find the coordinates of the point; if not, explain why.

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