
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 $x^3 - 4x^2 + 7x - 10$. Specifically, $t_0=3$, $t_1=4$, and $t_2=10$. Let $d$, $e$, and $f$ be real numbers such that $t_{k+1} = d \, t_k + e \, t_{k-1} + f \, t_{k-2}$ for $k = 2$, $3$, $....$ Determine the value of $d+e+f$.

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