<|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
 For an integer  $n\ge 3$  and a permutation  $\sigma=(p_{1},p_{2},\cdots ,p_{n})$  of  $\{1,2,\cdots , n\}$ , we say  $p_{l}$  is a  $landmark$  point if  $2\le l\le n-1$  and  $(p_{l-1}-p_{l})(p_{l+1}-p_{l})>0$ . For example , for  $n=7$ ,
the permutation  $(2,7,6,4,5,1,3)$  has four landmark points:  $p_{2}=7$ ,  $p_{4}=4$ ,  $p_{5}=5$  and  $p_{6}=1$ . For a given  $n\ge 3$  , let  $L(n)$  denote the number of permutations of  $\{1,2,\cdots ,n\}$  with exactly one landmark point. Find the maximum  $n\ge 3$  for which  $L(n)$  is a perfect square. 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
3
For an integer  $n\ge 3$  and a permutation  $\sigma=(p_{1},p_{2},\cdots ,p_{n})$  of  $\{1,2,\cdots , n\}$ , we say  $p_{l}$  is a  $landmark$  point if  $2\le l\le n-1$  and  $(p_{l-1}-p_{l})(p_{l+1}-p_{l})>0$ . For example , for  $n=7$ ,
the permutation  $(2,7,6,4,5,1,3)$  has four landmark points:  $p_{2}=7$ ,  $p_{4}=4$ ,  $p_{5}=5$  and  $p_{6}=1$ . For a given  $n\ge 3$  , let  $L(n)$  denote the number of permutations of  $\{1,2,\cdots ,n\}$  with exactly one landmark point. Find the maximum  $n\ge 3$  for which  $L(n)$  is a perfect square.
test