
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 hyperbola $C$: $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$ shares a common vertex with the hyperbola $\frac{x^{2}}{16} - \frac{y^{2}}{9} = 1$ and passes through the point $A(6, \sqrt{5})$.
1. Find the equation of the hyperbola $C$ and write out the equations of its asymptotes.
2. If point $P$ is a point on hyperbola $C$ and the distance from $P$ to the right focus is $6$, find the distance from $P$ to the left directrix.

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