
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 a parabola $y^2=8x$ with a focus at the point $F$, find the equation of a hyperbola $C: \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$ ($a>0, b>0$) such that the distance from the focus $F$ to the hyperbola's asymptote $ax - by = 0$ is $\frac{4\sqrt{5}}{5}$. Let point P be a moving point on the parabola $y^2 = 8x$, and let the sum of the distance from P to the focus $F$ of the hyperbola $C$ and to the line $x = -2$ be minimum, at a value of 3. Determine the equation of the hyperbola.

A) $\frac{y^2}{2} - \frac{x^2}{3} = 1$

B) $y^2 - \frac{x^2}{4} = 1$

C) $\frac{y^2}{4} - x^2 = 1$

D) $\frac{y^2}{3} - \frac{x^2}{2} = 1$

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