
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
Polynomial  $P(x)=c_{2006}x^{2006}+c_{2005}x^{2005}+\ldots+c_1x+c_0$  has roots  $r_1,r_2,\ldots,r_{2006}$ .  The coefficients satisfy  $2i\tfrac{c_i}{c_{2006}-i}=2j\tfrac{c_j}{c_{2006}-j}$  for all pairs of integers  $0\le i,j\le2006$ .  Given that  $\sum_{i\ne j,i=1,j=1}^{2006} \tfrac{r_i}{r_j}=42$ , determine  $\sum_{i=1}^{2006} (r_1+r_2+\ldots+r_{2006})$ .

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