
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
Find the rank of the \( (2n+1) \times (2n+1) \) skew-symmetric matrix with entries given by:
- \( a_{ij} = 1 \) for \( i - j = -2n, -(2n-1), \ldots, -(n+1) \)
- \( a_{ij} = -1 \) for \( i - j = -n, -(n-1), \ldots, -1 \)
- \( a_{ij} = 1 \) for \( i - j = 1, 2, \ldots, n \)
- \( a_{ij} = -1 \) for \( i - j = n+1, n+2, \ldots, 2n+1 \)

In other words, the main diagonal is zeros, the diagonals immediately below the main diagonal (up to \( n \) diagonals) are \( 1s \), the diagonals below those are \( -1s \), the diagonals immediately above the main diagonal (up to \( n \) diagonals) are \( -1s \), and the diagonals above those are \( 1s \).

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