
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
Given \( a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3} \in \mathbf{N} \) such that \( a_{1} + a_{2} + a_{3} = b_{1} + b_{2} + b_{3} \),
\[ 
a_{1}a_{2}a_{3} = b_{1}b_{2}b_{3} 
\]
\[ 
a_{1}a_{2} + a_{1}a_{3} + a_{2}a_{3} \neq b_{1}b_{2} + b_{1}b_{3} + b_{2}b_{3} 
\]

Then find the minimal value of the largest number among \( a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3} \).

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