
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 that $F_{1}$ and $F_{2}$ are the left and right foci of the hyperbola $C: \frac{x^{2}}{a^{2}} - \frac{y^{2}}{9} = 1 (a > 0)$, respectively, and point $P$ is any point on the hyperbola with $||PF_{1}| - |PF_{2}|| = 2$. The vertex is at the origin, and the parabola with the right vertex of the hyperbola as its focus is denoted as $L$.  
(Ⅰ) Find the equations of the asymptotes of the hyperbola $C$ and the standard equation of the parabola $L$;  
(Ⅱ) A line through the intersection point of the directrix of parabola $L$ and the x-axis intersects the parabola at points $M$ and $N$. What should be the slope of this line so that the circle with diameter $MN$ passes through the focus of parabola $L$?

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