
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
Find the minimum value of
\[
f(x) = x + \frac{2x}{x^2 + 1} + \frac{x(x + 3)}{x^2 + 2} + \frac{3(x + 1)}{x(x^2 + 2)}
\]
for \( x > 0 \).

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