
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
1. There are 5 students signing up for two extracurricular activity groups. Each student can only sign up for one group. How many different sign-up methods are there in total?
2. There is one coin each of 1 jiao, 5 jiao, 1 yuan, 5 yuan, 10 yuan, 50 yuan, and 100 yuan. How many different non-zero denominations can be formed?

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