
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 \( B = \{b_{1}, b_{2}, \cdots, b_{50} \} \), if a mapping \( f \) from \( A \) to \( B \) makes every element in \( B \) have a preimage, and
$$
f(a_{1}) \leq f(a_{2}) \leq \cdots \leq f(a_{100})
$$

then the number of such mappings is ( )
(A) \( \mathrm{C}_{100}^{50} \)
(C) \( \mathrm{C}_{100}^{49} \)
(D) \( \mathrm{C}_{99} \rightarrow 2+1 \)

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