
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
There are real numbers  $a,b$  and  $c$  and a positive number  $\lambda$  such that  $f(x)=x^3+ax^2+bx+c$  has three real roots  $x_1, x_2$  and  $x_3$  satisfying $(1) x_2-x_1=\lambda$  $(2) x_3>\frac{1}{2}(x_1+x_2)$ .
Find the maximum value of  $\frac{2a^3+27c-9ab}{\lambda^3}$ 

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