
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
Find all natural numbers  $ k\ge 1 $  and  $ n\ge 2, $  which have the property that there exist two matrices  $ A,B\in M_n\left(\mathbb{Z}\right) $  such that  $ A^3=O_n $  and  $ A^kB +BA=I_n. $ 

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