
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  $ n > 1$  be an odd positive integer and  $ A = (a_{ij})_{i, j = 1..n}$  be the  $ n \times n$  matrix with 
\[ a_{ij}= \begin{cases}2 & \text{if }i = j  1 & \text{if }i-j \equiv \pm 2 \pmod n  0 & \text{otherwise}\end{cases}.\]
Find  $ \det A$ .

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