
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
Assume \(a_{1}, a_{2}, \cdots, a_{n}\) is a permutation of the natural numbers \(1, 2, \cdots, n\). For all permutations of \(1, 2, \cdots, n\), find the maximum value of the sum
$$
\left|a_{1}-1\right|+\left|a_{2}-2\right|+\cdots+\left|a_{n}-n\right|.
$$

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