
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
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.

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