
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 the function \( f(x) \) is strictly decreasing on \( (0, +\infty) \) and for any \( x \in (0, +\infty) \), the equation \( f(x) \cdot f \left(f(x) + \frac{2}{x} \right) = \frac{1}{3} \) holds. Define \( g(x) = f(x) + 4x^2 \) for \( x \in (0, +\infty) \). Find the minimum value of the function \( g(x) \).

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