
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 lattice points $A\left(x_{1}, x_{2}, \cdots, x_{n}\right)$ and $B\left(y_{1}, y_{2}, \cdots, y_{n}\right)$ in $X$, their distance is defined as
$$
|A-B|=\left|x_{1}-y_{1}\right|+\left|x_{2}-y_{2}\right|+\cdots+\left|x_{n}-y_{n}\right| .
$$

Find the minimum value of $r$ such that it is possible to color the points in $X$ with $r$ colors, satisfying the following conditions:
(1) For any two lattice points $A$ and $B$ of the same color,
$$
|A-B| \geqslant 2 ;
$$
(2) For any two lattice points $A$ and $B$ of the same color,
$$
|A-B| \geqslant 3 .
$$

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