
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
A television station airs $r$ commercials over $n$ days. Determine:
(1) If at least $p$ commercials are aired each day $(n p \leq r)$, how many ways can the commercials be scheduled?
(2) If at most $q$ commercials are aired each day $(n q \geq r)$, how many ways can the commercials be scheduled?
(3) If the number of commercials aired each day is between $p$ and $q$ inclusive, how many ways can the commercials be scheduled?

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