
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 hyperbola $C$: $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ ($a > 0, b > 0$) with eccentricity $e= \frac{\sqrt{5}}{2}$, its right focus is $F$, and $O$ is the origin. The circle with diameter $OF$ intersects one of the asymptotes of the hyperbola $C$ at points $O$ and $A$. If the area of $\triangle AOF$ is $1$, determine the value of the real number $a$.

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