
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
There are 5 paths leading to the top of a mountain. Two people ascend and then descend. How many different ways can this happen if the two people are not distinguished and

a) each path can be used by at most one person and at most once,

b) each path can be used twice but only in different directions,

c) there are no restrictions,

d), e), and f) are the same as a), b), and c), but the two people are distinguished.

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