
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
Find the generating functions for the following combinatorial problems and determine the corresponding solutions:
(1) Find the $k$-combinations of $\{a_{1}, a_{2}, \cdots, a_{n}\}$;
(2) Find the $k$-combinations of $\{\infty \cdot a_{1}, \infty \cdot a_{2}, \cdots, \infty \cdot a_{n}\}$;
(3) Find the 10-combinations of $\{3 \cdot a, 4 \cdot b, 5 \cdot c\}$, where $\{3 \cdot a, 4 \cdot b, 5 \cdot c\}$ represents a set consisting of 3 $a$'s, 4 $b$'s, and 5 $c$'s.

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