
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 smallest positive integer \( n \) that is not less than 9, such that for any \( n \) integers (which can be the same) \( a_{1}, a_{2}, \cdots, a_{n} \), there always exist 9 numbers \( a_{i_{1}}, a_{i_{2}}, \cdots, a_{i_{9}} \) (where \(1 \leq i_{1} < i_{2} < \cdots < i_{9} \leq n \)) and \( b_{i} \in \{4,7\} \) (for \(i=1,2,\cdots,9\)), such that \( b_{1} a_{i_{1}} + b_{2} a_{i_{2}} + \cdots + b_{9} a_{i_{9}} \) is divisible by 9.

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