
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 given positive integer. Consider a permutation of  $1,2,3,\cdots,N$ , denoted as  $p_1,p_2,\cdots,p_N$ . For a section  $p_l, p_{l+1},\cdots, p_r$ , we call it "extreme" if  $p_l$  and  $p_r$  are the maximum and minimum value of that section. We say a permutation  $p_1,p_2,\cdots,p_N$  is "super balanced" if there isn't an "extreme" section with a length at least  $3$ . For example,  $1,4,2,3$  is "super balanced", but  $3,1,2,4$  isn't. Please answer the following questions:
1. How many "super balanced" permutations are there?
2. For each integer  $M\leq N$ . How many "super balanced" permutations are there such that  $p_1=M$ ?

*Proposed by ltf0501*

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