
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 $C\_1$: $x^{2}=2y$, and the hyperbola $C\_2$: $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 (a > 0, b > 0)$, its right vertex is $A$, and the eccentricity is $\sqrt{5}$. If the line passing through point $A$ and parallel to the asymptote of $C\_2$ is tangent to $C\_1$, then the standard equation of the hyperbola is $\_\_\_\_\_\_\_\_$.

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