
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 $t_k$ denote the sum of the $k$th powers of the roots of the polynomial $x^3 - 7x^2 + 12x - 20$. In particular, $t_0=3$, $t_1=7$, and $t_2=15$. Let $d$, $e$, and $f$ be real numbers such that $t_{k+1} = d t_k + e t_{k-1} + f t_{k-2} - 5$ for $k = 2$, $3$, ....$ What is $d+e+f$?
A) 12  
B) 13  
C) 14  
D) 15  
E) 17

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