
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 strictly positive integer. Roger has a rectangular garden of dimensions \( (2n + 1) \times (2n + 1) \), and he places barriers to divide his garden into several rectangular cells. Once finished, he will have exactly two horizontal rectangles \( k \times 1 \) and two vertical rectangles \( 1 \times k \) for each even integer \( k \) between 1 and \( 2n + 1 \), as well as a single \( 1 \times 1 \) square. How many ways does Roger have to do this?

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