
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 point $M(1,m)$ on the parabola $y^{2}=2px (p>0)$, where the distance from $M$ to its focus is $5$, and a hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1 (a>0, b>0)$ with its left vertex $A$ and eccentricity $\frac{\sqrt{5}}{2}$. If one of the asymptotes of the hyperbola is perpendicular to the line $AM$, find the equation of the hyperbola.

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