
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
Let  $F_m$  be the  $m$ 'th Fibonacci number, defined by  $F_1=F_2=1$  and  $F_m = F_{m-1}+F_{m-2}$  for all  $m \geq 3$ . Let  $p(x)$  be the polynomial of degree 1008 such that  $p(2n+1)=F_{2n+1}$  for  $n=0,1,2,\ldots,1008$ . Find integers  $j$  and  $k$  such that  $p(2019) = F_j - F_k$ .


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