
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 vacuum tube has 5 terminals and its socket has 5 holes. These terminals are located at the vertices of a regular pentagon, and the tube can be inserted into the socket in 5 different ways. The terminals are numbered from 1 to 5. In how many ways can the holes be numbered from 1 to 5 such that in every possible position of the tube, at least one terminal is in the hole with the same number?

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