
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 any \( m \) distinct numbers selected from \( 1, 2, \cdots, n \). If there exists \( k \in \{1, 2, \cdots, m\} \) such that \( a_k + k \) is odd, or there exist integers \( k \) and \( l \) (with \( 1 \leq k < l \leq m \)) such that \( a_k > a_l \), then \( a_1, a_2, \cdots, a_m \) is called a "good permutation". Determine the number of all good permutations.

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