
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 \( A = \{a_{1}, a_{2}, \cdots, a_{100}\} \) and \( B = \{ b_{1}, b_{2}, \cdots, b_{50} \} \), with a mapping \( f \) from \( A \) to \( B \) such that every element in \( B \) is mapped to by some element in \( A \), and \( f(a_{1}) \leq f(a_{2}) \leq \cdots \leq f(a_{100}) \), determine the number of such mappings.

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