
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 \(\left(a_{1}, a_{2}, \ldots, a_{100}\right)\) be a permutation of \(\{1,2, \ldots, 100\}\). Define \(L(k)\) as the largest \(m\) such that there exists \(k=i_{1}<i_{2}<\ldots<i_{m}\) where either \(a_{i_{1}}>a_{i_{2}}>\ldots>a_{i_{m}}\) or \(a_{i_{1}}<a_{i_{2}}<\ldots<a_{i_{m}}\). Find the minimum value of \(\sum_{k=1}^{100} L(k)\).

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