
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 the function \( f(x, y) = 6\left(x^{2} + y^{2}\right)(x + y) - 4\left(x^{2} + xy + y^{2}\right) - 3(x + y) + 5 \) in the region \( D = \{(x, y) \mid x > 0, y > 0\} \).

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