
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 $A=\{a_1, a_2, a_3, a_4\}$ and $B=\{0, 1, 2, 3\}$. Let $f$ be a mapping from $A$ to $B$.
1. If every element in $B$ has a preimage, how many different mappings $f$ exist?
2. If $f$ satisfies $f(a_1) + f(a_2) + f(a_3) + f(a_4) = 4$, how many different mappings $f$ exist?

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