
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 total of 35 commodities in a city's commerce bureau were sampled for inspection, among which 15 were counterfeit. Select 3 commodities from these 35.

(1) How many different ways are there if one of the counterfeit commodities must be included?
(2) How many different ways are there if one of the counterfeit commodities cannot be included?
(3) How many different ways are there if exactly 2 counterfeit commodities are included?
(4) How many different ways are there if at least 2 counterfeit commodities are included?
(5) How many different ways are there if at most 2 counterfeit commodities are included?

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