
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 \( A=\left\{a_{1}, a_{2}, \cdots, a_{100}\right\} \) and \( B=\left\{b_{1}, b_{2}, \cdots, b_{50}\right\} \), consider a mapping \( f \) from \( A \) to \( B \) such that every element in \( B \) has a preimage and \( f\left(a_{1}\right) \leqslant f\left(a_{2}\right) \leqslant \cdots \leqslant f\left(a_{100}\right) \). How many such mappings exist?

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