
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
(1) Point $F$ is the lower focus of the ellipse $\left({{x}^{2}}+\dfrac{{{y}^{2}}}{{{a}^{2}}}=1\right)$, and points $A$ and $B$ are its vertices on the positive semi-axes of the $x$-axis and $y$-axis, respectively. Point $P$ is on the ellipse, $PF \perp y$-axis, and $OP$ is parallel to $AB$. Find the equation of the ellipse.

(2) The asymptotes of a hyperbola are given by the equations $y=\pm \dfrac{1}{2}x$, and the distance between its two vertices is $4$. Find the standard equation of this hyperbola.

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