
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 $\dfrac{y^{2}}{a^{2}} - \dfrac{x^{2}}{b^{2}} = 1$ $(a > 0, b > 0)$, one of its asymptote equations is $2x + y = 0$, and the distance from the vertex to the asymptote is $\dfrac{2\sqrt{5}}{5}$.

$(1)$ Find the equation of this hyperbola;

$(2)$ Let $P$ be a point on the hyperbola, and points $A$, $B$ are on the asymptotes of the hyperbola, located in the first and second quadrants, respectively. If $\vec{AP} = \vec{PB}$, find the area of $\triangle AOB$.

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