
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
How many positive roots does polynomial  $x^{2002} + a_{2001}x^{2001} + a_{2000}x^{2000} + \cdots + a_1x + a_0$  have such that  $a_{2001} = 2002$  and  $a_k = -k - 1$  for  $0\leq k \leq 2000$ ? $ 
\textbf{a)}\ 0
\qquad\textbf{b)}\ 1
\qquad\textbf{c)}\ 2
\qquad\textbf{d)}\ 1001
\qquad\textbf{e)}\ 2002
$ 

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