
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 $x^3 - 5x^2 + 6x - 7$. We know that $t_0 = 3$, $t_1 = 5$, and $t_2 = 9$. Let $\alpha$, $\beta$, and $\gamma$ be real numbers such that for $k = 2, 3, ...$, the relation $t_{k+1} = \alpha t_k + \beta t_{k-1} + \gamma t_{k-2} + 2$ holds. Determine the value of $\alpha + \beta + \gamma$.

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