
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
Alice, Bob, and Carol each independently roll a fair six-sided die and obtain the numbers  $a, b, c$ , respectively. They then compute the polynomial  $f(x)=x^{3}+p x^{2}+q x+r$  with roots  $a, b, c$ . If the expected value of the sum of the squares of the coefficients of  $f(x)$  is  $\frac{m}{n}$  for relatively prime positive integers  $m, n$ , find the remainder when  $m+n$  is divided by 1000 .

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