
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 \( O \) is the origin of coordinates, the functions 
\[ f(x) = x^2 \text{ (for } x > 0) \] 
and 
\[ g(x) = -\frac{1}{x} \text{ (for } x > 0) \text{,} \]
points \( A \) and \( B \) lie on the graphs of \( f(x) \) and \( g(x) \) respectively, such that \( \angle AOB = 120^\circ \). Find the minimum area of triangle \( \triangle AOB \) (accurate to \( 10^{-4} \)).

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