
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 parabola $y^{2}=8x$ and its focus $F$ is at a distance of $\frac{4\sqrt{5}}{5}$ from the asymptote of the hyperbola $\frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}}=1,(a>0,b>0)$. Let $P$ be a moving point on the parabola $y^{2}=8x$, the minimum value of the sum of the distance from $P$ to the upper focus $F_1(0,c)$ of the hyperbola and the distance to the line $x+2=0$ is $3$. Determine the equation of the hyperbola $\frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}}=1$.

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