
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 \( m, n \in \mathbf{N}_{+} \), with \( m \leq n \). Define the sets \( A = \{ a_1, \ldots, a_m \} \) and \( B = \{ b_1, \ldots, b_n \} \).

1. Find the number of all mappings from \( A \) to \( B \).
2. Find the number of all injective mappings from \( A \) to \( B \).
3. Determine whether there exists a surjective mapping from \( A \) to \( B \).

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