
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 \( x, y > 0 \), if
\[ f(x, y) = \left( x^2 + y^2 + 2 \right) \left( \frac{1}{x+y} + \frac{1}{xy+1} \right), \]
then the minimum value of \( f(x, y) \) is \(\quad\).

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