
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 certain city's Industry and Commerce Bureau conducted a sampling inspection of 35 types of goods, identifying 15 types as counterfeit. Now, 3 types of goods are selected from these 35 types.  
(1) If one specific counterfeit must be included, how many different selections are there?  
(2) If one specific counterfeit must not be included, how many different selections are there?  
(3) If exactly 2 counterfeits are included, how many different selections are there?  
(4) If at least 2 counterfeits are included, how many different selections are there?  
(5) If at most 2 counterfeits are included, how many different selections are there?

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