
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 \( n \) be a natural number. Roger has a square garden of size \( (2n + 1) \times (2n + 1) \). He erects fences to divide it into rectangular beds. He wants exactly two horizontal \( k \times 1 \) beds and exactly two vertical \( 1 \times k \) beds for every even number \( k \) between 1 and \( 2n + 1 \), as well as a single square bed of size \( 1 \times 1 \) when he is finished. How many different ways can Roger divide his garden?

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