
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 TV station aired $r$ commercials over $n$ days. 

1. If at least $p$ commercials are aired each day $(n p \leqslant r)$, how many ways can the commercials be aired?
2. If at most $q$ commercials are aired each day $(n q \geqslant r)$, how many ways can the commercials be aired?
3. If the number of commercials aired each day is between $[p, q]$, how many ways can the commercials be aired?

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