
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
We are given some three element subsets of  $\{1,2, \dots ,n\}$  for which any two of them have at most one common element. We call a subset of  $\{1,2, \dots ,n\}$  *nice* if it doesn't include any of the given subsets. If no matter how the three element subsets are selected in the beginning, we can add one more element to every 29-element *nice* subset while keeping it nice, find the minimum value of  $n$ .

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