
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
$$
X=\left\{\left(x_{1}, x_{2}, \cdots, x_{n}\right) \mid x_{1}, x_{2}, \cdots, x_{n} \in \mathbf{Z}\right\}
$$
be the set of all lattice points in $n$-dimensional space. For any two points $A\left(x_{1}, x_{2}, \cdots, x_{n}\right)$ and $B\left(y_{1}, y_{2}, \cdots, y_{n}\right)$ in $X$, define their distance as
$$
|A-B|=\left|x_{1}-y_{1}\right|+\left|x_{2}-y_{2}\right|+\cdots+\left|x_{n}-y_{n}\right|.
$$

The set $Q(A)=\{P \in X \mid |P-A| \leqslant 1\}$ is called a set centered at $A$.

If some, but not all, points in $X$ can be colored red such that each set $Q(A)$ $(A \in X)$ contains exactly $k$ red points, find all possible values of the positive integer $k$.

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