
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  $\mathcal M$  be the set of  $n\times n$  matrices with integer entries. Find all  $A\in\mathcal M$  such that  $\det(A+B)+\det(B)$  is even for all  $B\in\mathcal M$ .

*Proposed by Ethan Tan*

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