
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
Given an integer \( n \geq 2 \), consider a permutation of \(\{1, 2, \ldots, n\}\) denoted as \(\left(a_{1}, a_{2}, \ldots, a_{n}\right)\). For \( 1 \leq i \leq n \), let \( x_{i} \) represent the maximum length of the increasing subsequence starting with \( a_{i} \), and let \( y_{i} \) represent the maximum length of the decreasing subsequence starting with \( a_{i} \). Determine the minimum possible value of \(\sum_{\mathrm{i}=1}^{\mathrm{n}}\left|x_{i}-y_{i}\right|\).

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