
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  $S$  be a planar region. A  $\emph{domino-tiling}$  of  $S$  is a partition of  $S$  into  $1\times2$  rectangles. (For example, a  $2\times3$  rectangle has exactly  $3$  domino-tilings, as shown below.)

[asy]
import graph; size(7cm); 
pen dps = linewidth(0.7); defaultpen(dps);

draw((0,0)--(3,0)--(3,2)--(0,2)--cycle, linewidth(2)); 
draw((4,0)--(4,2)--(7,2)--(7,0)--cycle, linewidth(2)); 
draw((8,0)--(8,2)--(11,2)--(11,0)--cycle, linewidth(2)); 

draw((1,0)--(1,2));
draw((2,1)--(3,1));
draw((0,1)--(2,1), linewidth(2));
draw((2,0)--(2,2), linewidth(2));

draw((4,1)--(7,1));
draw((5,0)--(5,2), linewidth(2));
draw((6,0)--(6,2), linewidth(2));

draw((8,1)--(9,1));
draw((10,0)--(10,2));
draw((9,0)--(9,2), linewidth(2));
draw((9,1)--(11,1), linewidth(2));

[/asy]
The rectangles in the partition of  $S$  are called  $\emph{dominoes}$ .
(a) For any given positive integer  $n$ , find a region  $S_n$  with area at most  $2n$  that has exactly  $n$  domino-tilings.

(b) Find a region  $T$  with area less than  $50000$  that has exactly  $100002013$  domino-tilings.

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