
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 \( s_k \) denote the sum of the \( k \)-th powers of the roots of the polynomial \( x^3 - 7x^2 + 12x - 17 \). Specifically, \( s_0 = 3 \), \( s_1 = 7 \), and \( s_2 = 17 \). Let \( a, b, \) and \( c \) be real numbers such that \( s_{k+1} = a s_k + b s_{k-1} + c s_{k-2} \) for \( k = 2, 3, \ldots \). What is \( a + b + c \)?
A) 10
B) 11
C) 12
D) 13
E) 14

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