
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 $F_1$ 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, point P is any point on the hyperbola, and $||PF_1|-|PF_2||=2$. 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 equation of the asymptotes of the hyperbola C and the standard equation of the parabola L;  
(Ⅱ) A line through the intersection of the directrix of parabola L and the x-axis intersects the parabola at points M and N. What should be the slope of the 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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