
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 positive integer  $n$ , find the largest real number  $\lambda=\lambda(n)$ , such that for any degree  $n$  polynomial with complex coefficients  $f(x)=a_n x^n+a_{n-1} x^{n-1}+\cdots+a_0$ ,
and any permutation  $x_0,x_1,\cdots,x_n$  of  $0,1,\cdots,n$ , the following inequality holds  $\sum_{k=0}^n|f(x_k)-f(x_{k+1})|\geq \lambda |a_n|$ , where  $x_{n+1}=x_0$ .

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