
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 $k$ be a positive integer, $A \subseteq \mathbf{Z}$. If there exist $x_{1}, x_{2}, \cdots, x_{k} \in \mathbf{Z}$ such that for all $i$ and $j$ with $1 \leq i < j \leq k$, we have $\left(x_{i} + A\right) \cap \left(x_{j} + A\right) = \varnothing$, then $A$ is called a $k$-set, where $x + A = \{x + a \mid a \in A\}$.

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