
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 plane $(xOy)$, an ellipse $(C)$ is given by the equation: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with an eccentricity of $\frac{\sqrt{6}}{3}$ and passes through the point $(3,-1)$.

1. Find the semi-axes of the ellipse $(C)$.
2. If a moving point $P$ is on the line $l$: $x = -2\sqrt{2}$, draw a line through $P$ intersecting the ellipse $(C)$ at points $M$ and $N$ such that $PM = PN$. Then, draw a line $l' \perp MN$ through $P$. Determine whether $l'$ always passes through a fixed point. If so, find the coordinates of this point. If not, explain why.

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