
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 two sets of real numbers $A=\{a_1, a_2, \ldots, a_{100}\}$ and $B=\{b_1, b_2, \ldots, b_{50}\}$. If there is a mapping $f$ from $A$ to $B$ such that every element in $B$ has a pre-image in $A$, and $f(a_1) \leq f(a_2) \leq \ldots \leq f(a_{100})$, how many such mappings are there? (Answer with symbols).

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