
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
 $M$  and  $N$  are real unequal  $n\times n$  matrices satisfying  $M^3=N^3$  and  $M^2N=N^2M$ . Can we choose  $M$  and  $N$  so that  $M^2+N^2$  is invertible?

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