
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  $A$  be a  $2n\times 2n$  matrix, with entries chosen independently at random. Every entry is chosen to be  $0$  or  $1,$  each with probability  $1/2.$  Find the expected value of  $\det(A-A^t)$  (as a function of  $n$ ), where  $A^t$  is the transpose of  $A.$ 

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