
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
Given a positive integer $k \geq 2$, if we select $k$ vertices from the cube $ABCD-A_1B_1C_1D_1$ to form a set $M=\{X_1, X_2, \ldots, X_k\}$, satisfying $\forall X_i, X_j \in M$, there exist $X_l, X_t \in M$ such that the line $X_iX_j \perp X_lX_t$, then the possible values of $k$ are \_\_\_\_\_\_.

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