
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
Team A and Team B each have $n$ members. It is known that each member of Team A shakes hands with each member of Team B exactly once (members within the same team do not shake hands). From these $n^2$ handshakes, two are randomly selected. Let event A be the event that exactly 3 members are involved in these two handshakes. If the probability of event A happening, $P$, is less than $\frac{1}{10}$, then the minimum value of $n$ is \_\_\_\_\_\_.

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