
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
In a gathering of $40$ people, there are $30$ people who all know each other and $10$ people, of whom $3$ know exactly $5$ members of the first group, while the other $7$ know none. Find the number of handshakes that occur within the group.

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