
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
If a permutation \(\left\{i_{1}, i_{2}, \cdots, i_{n}\right\}\) of \(\{1,2, \cdots, n\}\) satisfies \(i_{1} \neq 1, i_{2} \neq 2, \cdots, i_{n} \neq n\), then \(\left\{i_{1}, i_{2}, \cdots, i_{n}\right\}\) is called a derangement of \(\{1,2, \cdots, n\}\). Find the total number of derangements \(D_{n}\) of \(\{1,2, \cdots, n\}\).

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