
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 say three sets  $A_1, A_2, A_3$  form a triangle if for each  $1 \leq i,j \leq 3$  we have  $A_i \cap A_j \neq \emptyset$ , and  $A_1 \cap A_2 \cap A_3 = \emptyset$ . Let  $f(n)$  be the smallest positive integer such that any subset of  $\{1,2,3,\ldots, n\}$  of the size  $f(n)$  has at least one triangle. Find a formula for  $f(n)$ .

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