
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 $n \geq 9$ such that for any $n$ integers (which may be the same) $a_1, a_2, \cdots, a_n$, there always exist 9 numbers $a_{i_1}, a_{i_2}, \cdots, a_{i_9} (1 \leq i_1 < i_2 < \cdots < i_9 \leq n)$ and $b_i \in \{4, 7\} (i=1, 2, \cdots, 9)$ 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