
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 $k\geqslant 2$, if $k$ vertices are chosen from the $8$ vertices of a cube $ABCD-A_1B_1C_1D_1$ to form a set $M=\{X_1, X_2, ..., X_k\}$ such that for every pair of vertices $X_i, X_j \in M$, there exists another pair of vertices $X_l, X_t \in M$ for which the line segment $X_iX_j$ is perpendicular to line segment $X_lX_t$, then all possible values of $k$ are _________.

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