
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
Find the largest positive integer \( n \) for which there exist \( n \) finite sets \( X_{1}, X_{2}, \ldots, X_{n} \) with the property that for every \( 1 \leq a < b < c \leq n \), the equation
\[ 
\left|X_{a} \cup X_{b} \cup X_{c}\right| = \lceil\sqrt{a b c}\rceil 
\]
holds.

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