
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
Define the even function $f(x-2)$ on $\mathbb{R}$. When $x > -2$, $f(x) = e^{x+1} - 2$. If there exists $k \in \mathbb{Z}$ such that the real root $x$ of the equation $f(x) = 0$ is in the interval $(k-1, k)$, determine the set of possible values for $k$.

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