
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 \(a_{1}, a_{2}, \ldots, a_{n}\) represent the natural numbers \(1, 2, \ldots, n\) arranged in some order. Determine the maximum value of the sum

\[ 
\left|a_{1} - 1\right| + \left|a_{2} - 2\right| + \ldots + \left|a_{n} - n\right| 
\]

considering all possible arrangements.

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