
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 \( a, b, x, y \) are non-negative real numbers and \( a + b = 27 \), determine the maximum value of \(\lambda\) such that the inequality \((a x^{2} + b y^{2} + 4 x y)^{3} \geqslant \lambda (a x^{2} y + b x y^{2})^{2}\) always holds. Also, find the conditions under which equality is achieved.

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