
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
Denote by  $F_0(x)$ ,  $F_1(x)$ ,  $\ldots$  the sequence of Fibonacci polynomials, which satisfy the recurrence  $F_0(x)=1$ ,  $F_1(x)=x$ , and  $F_n(x)=xF_{n-1}(x)+F_{n-2}(x)$  for all  $n\geq 2$ . It is given that there exist unique integers  $\lambda_0$ ,  $\lambda_1$ ,  $\ldots$ ,  $\lambda_{1000}$  such that \[x^{1000}=\sum_{i=0}^{1000}\lambda_iF_i(x)\] for all real  $x$ .  For which integer  $k$  is  $|\lambda_k|$  maximized?

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