
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 $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ $(a, b > 0)$ with foci $F\_1$ and $F\_2$, and a point $P$ on the ellipse such that $|PF\_1| = \frac{4}{3}$ and $|PF\_2| = \frac{14}{3}$, with $PF\_1 \perp F\_1F\_2$.

(I) Find the equation of the ellipse $C$.

(II) If a line $L$ passes through the center $M$ of the circle $x^2 + y^2 + 4x - 2y = 0$, intersecting the ellipse at points $A$ and $B$, and $A$ and $B$ are symmetric about point $M$, find the equation of line $L$.

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