
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  $a_1, a_2, \cdots a_{100}$  be a permutation of  $1,2,\cdots 100$ .
Define  $l(k)$  as the maximum  $m$  such that there exists  $i_1, i_2 \cdots i_m$  such that  $a_{i_1} > a_{i_2} > \cdots > a_{i_m}$  or  $a_{i_1} < a_{i_2} < \cdots < a_{i_m}$ , where  $i_1=k$  and  $i_1<i_2< \cdots <i_m$ Find the minimum possible value for  $\sum_{i=1}^{100} l(i)$ .



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