
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 $\textit{k}$th powers of the roots of the polynomial $x^3 - 4x^2 + 6x - 24$. Specifically, $t_0=3$, $t_1=4$, and $t_2=10$. Let $x$, $y$, and $z$ be real numbers such that $t_{k+1} = x t_k + y t_{k-1} + z t_{k-2}$ for $k = 2$, $3$, $...$. What is $x+y+z$?
A) 12
B) 18 
C) 22 
D) 28 
E) 34

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