
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
Real numbers \(a\), \(b\), and \(c\) and positive number \(\lambda\) make \(f(x) = x^3 + ax^2 + b x + c\) have three real roots \(x_1\), \(x_2\), \(x_3\), such that:
(1) \(x_2 - x_1 = \lambda\);
(2) \(x_3 > \frac{1}{2}(x_1 + x_2)\).
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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