
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 be two matrices  $ A,B\in\mathcal{M}_2\left( \mathbb{R} \right) $  that don’t commute.**a)** If  $ A^3=B^3, $  then  $ \text{tr} \left( A^n \right) =\text{tr} \left( B^n \right) ,  $  for all natural numbers  $ n. $ **b)** If  $ A^n\neq B^n $  and  $ \text{tr} \left( A^n \right) =\text{tr} \left( B^n \right) , $  for all natural numbers  $ n, $  then find some of the matrices  $ A,B. $ 

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