
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
The ellipse $C$: $\dfrac{x^{2}}{a^{2}} + \dfrac{y^{2}}{b^{2}} = 1 \ (a > b > 0)$ has two foci $F_{1}$ and $F_{2}$. A point $P$ is on the ellipse $C$, and $PF_{1} \perpendicular PF_{2}$, $|PF_{1}|= \dfrac{4}{3}$, $|PF_{2}|= \dfrac{14}{3}$,

 $(1)$ Find the equation of the ellipse.  
 $(2)$ If a line $L$ passes through the center $M$ of the circle $x^{2}+y^{2}+4x-2y=0$ and intersects the ellipse $C$ at points $A$ and $B$, and $A$, $B$ are symmetric about point $M$, find the equation of line $L$.

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