
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. Let  $S$  be the set of  $n^2$  cells in an  $n\times n$  grid. Call a subset  $T$  of  $S$  a **double staircase** if


- $T$  can be partitioned into  $n$  horizontal nonoverlapping rectangles of dimensions  $1 \times 1,
1 \times 2, ..., 1 \times n,$  and
- $T$  can also be partitioned into  $n$  vertical nonoverlapping rectangles of dimensions  $1\times1,
2 \times 1, ..., n \times 1$ .

In terms of  $n$ , how many double staircases are there? (Rotations and reflections are considered distinct.)
An example of a double staircase when  $n = 3$  is shown below.

[asy]
unitsize(1cm);
for (int i = 0; i <= 3; ++i)
{
draw((0,i)--(3,i),linewidth(0.2));
draw((i,0)--(i,3),linewidth(0.2));
}
filldraw((0,0)--(1,0)--(1,1)--(0,1)--cycle, lightgray, linewidth(0.2));
filldraw((1,0)--(2,0)--(2,1)--(1,1)--cycle, lightgray, linewidth(0.2));
filldraw((2,0)--(3,0)--(3,1)--(2,1)--cycle, lightgray, linewidth(0.2));
filldraw((0,1)--(1,1)--(1,2)--(0,2)--cycle, lightgray, linewidth(0.2));
filldraw((1,1)--(2,1)--(2,2)--(1,2)--cycle, lightgray, linewidth(0.2));
filldraw((1,2)--(2,2)--(2,3)--(1,3)--cycle, lightgray, linewidth(0.2));
[/asy]

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