
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
Consider a regular cube with side length  $2$ .  Let  $A$  and  $B$  be  $2$  vertices that are furthest apart.  Construct a sequence of points on the surface of the cube  $A_1$ ,  $A_2$ ,  $\ldots$ ,  $A_k$  so that  $A_1=A$ ,  $A_k=B$  and for any  $i = 1,\ldots, k-1$ , the distance from  $A_i$  to  $A_{i+1}$  is  $3$ .  Find the minimum value of  $k$ .

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