
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 positive integers \( m \) and \( n \) such that \( 2 \leq m \leq n \), let \( a_{1}, a_{2}, \cdots, a_{m} \) be a permutation of \( m \) distinct numbers chosen from \( 1, 2, \cdots, n \). This permutation is called a "good permutation" if there exists an integer \( k \) in \(\{1, 2, \cdots, m\}\) such that \( a_{k} + k \) is odd, or there exist integers \( k \) and \( l \) (\(1 \leq k < l \leq m\)) such that \( a_{k} > a_{l} \). Determine the number of all good permutations.

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