<|im_start|>system
You are a helpful assistant. You first think about the reasoning process in the mind and then provide the user with the answer. You should try to truely solve the question by yourself without cheating.  <|im_end|>
<|im_start|>user
 
A $2 \times 2$ square can be cut into a small square and an L-shaped piece (corner) in four ways. All ways of cutting are shown in the picture below.

How many ways are there to cut a $4 \times 4$ square into three L-shaped pieces (corners) and a small square, as depicted in the picture? (Methods that differ by rotation or reflection of the square are considered distinct.) Show your work in <think> </think> tags. And return the final answer in <answer> </answer> tags.<|im_end|>
<|im_start|>assistant
Let me solve this step by step.
<think>
user
64

A $2 \times 2$ square can be cut into a small square and an L-shaped piece (corner) in four ways. All ways of cutting are shown in the picture below.

How many ways are there to cut a $4 \times 4$ square into three L-shaped pieces (corners) and a small square, as depicted in the picture? (Methods that differ by rotation or reflection of the square are considered distinct.)
test