
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 \(\pi = a_1 a_2 \cdots a_n\) be any permutation of \(1, 2, \cdots, n\). If \(a_i = i \) for \(1 \leq i \leq n\), then \(i\) is said to retain its position in the permutation \(\pi\). Let \(D_n\) denote the number of permutations of \(1, 2, \cdots, n\) where no number retains its position. Find the formula to count \(D_n\).

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