
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 parabola ${x}^{2}=2py(p>0)$ and a hyperbola $\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1(a>0,b>0)$, determine the value of $p$ such that the directrix of the parabola and the asymptotes of the hyperbola form an isosceles right triangle with an area of $1$.

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