
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 real numbers \(a, b, c\) and a positive number \(\lambda\), such that the cubic polynomial \(f(x) = x^3 + a x^2 + b x + c\) has three real roots \(x_1, x_2, x_3\), and the following conditions hold:
(i) \( x_2 - x_1 = \lambda \);
(ii) \( x_3 > \frac{1}{2} (x_1 + x_2) \).

Find the maximum value of \( \frac{2 a^3 + 27 c - 9 a b}{\lambda^3} \).

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