
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 that point $A(m,1)$ is a point on the parabola $C:y^{2}=2px\ (p>0)$, a perpendicular line is drawn from point $A$ to the directrix, with the foot of the perpendicular being point $B$. If the area of $\triangle AOB$ (where $O$ is the origin) is $\frac{1}{2}$, calculate the value of $p$.

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