
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
In the Cartesian coordinate system $xOy$, an ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ has an eccentricity of $\frac{\sqrt{6}}{3}$ and passes through the point $(3, -1)$.

(1) Find the semi-latus rectum of ellipse $C$.

(2) A moving point $P$ is on the line $l: x = -2\sqrt{2}$. A line passing through $P$ intersects the ellipse $C$ at points $M$ and $N$, such that $PM = PN$. Then, a line $l'$ is drawn through $P$ perpendicular to $MN$. Determine whether $l'$ always passes through a fixed point. If so, find the coordinates of that point. If not, explain why.

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