
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
For a positive integer  $n$ , consider a square cake which is divided into  $n \times n$  pieces with at most one strawberry on each piece. We say that such a cake is *delicious* if both diagonals are fully occupied, and each row and each column has an odd number of strawberries.
Find all positive integers  $n$  such that there is an  $n \times n$  delicious cake with exactly  $\left\lceil\frac{n^2}{2}\right\rceil$  strawberries on it.

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