
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 positive integer. Roger has a \( (2n+1) \times (2n+1) \) square garden. He puts down fences to divide his garden into rectangular plots. He wants to end up with exactly two horizontal \( k \times 1 \) plots and exactly two vertical \( 1 \times k \) plots for each even integer \( k \) between 1 and \( 2n+1 \), as well as a single \( 1 \times 1 \) square plot. How many different ways are there for Roger to do this?

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